The document summarizes the formulation of a coupled Stokes-Darcy problem. It introduces the continuous problem, rewrites it using additional unknowns, defines function spaces for the unknowns, and presents the variational formulation consisting of equations for the Stokes and Darcy subproblems and coupling terms. It also decomposes the stress tensor for the Stokes problem. The problem models fluid flow between a fluid domain and porous medium.
1) The document discusses soliton scattering amplitudes in affine Toda field theory and how they can be calculated using quantum affine algebras and Yangians.
2) It presents an algebraic method to find solutions to the reflection equation which describes particle reflection at boundaries in the quantum theory.
3) The goal is to motivate studying certain coideal subalgebras of quantum affine algebras and Yangians in order to calculate reflection matrices and learn about boundary solitons.
The document discusses using a run test to determine if a sequence is random or not. A run is defined as a sequence of identical symbols or events. The number of runs in a random sequence will follow a normal distribution. For the given sequence of acceptable and damaged glass sculptures, the number of runs is calculated and compared to the expected number of runs for a random sequence to test the randomness of damage at the 0.05 significance level. The number of runs observed is found to not be significantly different than expected, so the damage is determined to occur randomly.
Computation of the gravity gradient tensor due to topographic masses using te...Leonardo Uieda
The GOCE satellite mission has the objective of measuring the Earth's gravitational field with an unprecedented accuracy through the measurement of the gravity gradient tensor (GGT). One of the several applications of this new gravity data set is to study the geodynamics of the lithospheric plates, where the flat Earth approximation may not be ideal and the Earth's curvature should be taken into account. In such a case, the Earth could be modeled using tesseroids, also called spherical prisms, instead of the conventional rectangular prisms. The GGT due to a tesseroid is calculated using numerical integration methods, such as the Gauss-Legendre Quadrature (GLQ), as already proposed by Asgharzadeh et al. (2007) and Wild-Pfeiffer (2008). We present a computer program for the direct computation of the GGT caused by a tesseroid using the GLQ. The accuracy of this implementation was evaluated by comparing its results with the result of analytical formulas for the special case of a spherical cap with computation point located at one of the poles. The GGT due to the topographic masses of the Parana basin (SE Brazil) was estimated at 260 km altitude in an attempt to quantify this effect on the GOCE gravity data. The digital elevation model ETOPO1 (Amante and Eakins, 2009) between 40º W and 65º W and 10º S and 35º S, which includes the Paraná Basin, was used to generate a tesseroid model of the topography with grid spacing of 10' x 10' and a constant density of 2670 kg/m3. The largest amplitude observed was on the second vertical derivative component (-0.05 to 1.20 Eötvos) in regions of rough topography, such as that along the eastern Brazilian continental margins. These results indicate that the GGT due to topographic masses may have amplitudes of the same order of magnitude as the GGT due to density anomalies within the crust and mantle.
This document is a master's thesis written in Chinese that investigates the existence and uniqueness of solutions to stochastic differential equations (SDEs) with Lévy noise and non-Lipschitz coefficients. It introduces Lévy processes and their properties, including the Lévy-Itô decomposition. It defines stochastic integration with respect to compensated Poisson processes and provides Itô's formula for Lévy diffusions. The thesis proves that if weak existence and pathwise uniqueness hold for an SDE with Lévy noise, then it has a unique strong solution. It establishes conditions on the coefficients that ensure infinite lifetime and pathwise uniqueness of the solution.
This document discusses relations, functions, and matrices. It defines binary relations as subsets of the Cartesian product of a set with itself that satisfy a given relationship. It discusses properties of relations such as reflexive, symmetric, transitive, and antisymmetric relations. It also defines partial orderings, equivalence relations, and partitioning sets into equivalence classes. Functions are introduced as a special type of binary relation with unique outputs. Matrices are defined as a way to represent linear functions.
Solvability of Matrix Riccati Inequality Talk SlidesKevin Kissi
15min presentation slides. It goes beyond Beamer latex to showcase the best use of color and design in a Mathematical talk slide.
The paper itself is archived at arxiv.org with pdf at: https://arxiv.org/pdf/1505.04861.pdf
This document discusses the breakdown of linearization in the Vlasov equation due to particle trapping in an oscillating electric field. It shows that the linear Landau damping solution is only valid when the perturbation amplitude is small and for times less than the trapping time scale τtr. Beyond τtr, particles become trapped in oscillation in the electric potential well, violating the assumption of unperturbed trajectories used in the linearized solution. Nonlinear effects then dominate and nonlinear methods are required.
This document is a calculus supplement to accompany a microeconomics textbook. It introduces the concept of partial derivatives and shows how they can be used to analyze economic concepts from the textbook like demand and supply functions, substitutes and complements, and elasticities. Partial derivatives allow the slope of demand and supply functions to be determined with respect to different variables. They also allow elasticities to be defined and calculated using calculus, providing an equivalent but alternative method to the algebraic approach in the textbook. The supplement aims to illustrate the connections between calculus and microeconomic concepts.
1) The document discusses soliton scattering amplitudes in affine Toda field theory and how they can be calculated using quantum affine algebras and Yangians.
2) It presents an algebraic method to find solutions to the reflection equation which describes particle reflection at boundaries in the quantum theory.
3) The goal is to motivate studying certain coideal subalgebras of quantum affine algebras and Yangians in order to calculate reflection matrices and learn about boundary solitons.
The document discusses using a run test to determine if a sequence is random or not. A run is defined as a sequence of identical symbols or events. The number of runs in a random sequence will follow a normal distribution. For the given sequence of acceptable and damaged glass sculptures, the number of runs is calculated and compared to the expected number of runs for a random sequence to test the randomness of damage at the 0.05 significance level. The number of runs observed is found to not be significantly different than expected, so the damage is determined to occur randomly.
Computation of the gravity gradient tensor due to topographic masses using te...Leonardo Uieda
The GOCE satellite mission has the objective of measuring the Earth's gravitational field with an unprecedented accuracy through the measurement of the gravity gradient tensor (GGT). One of the several applications of this new gravity data set is to study the geodynamics of the lithospheric plates, where the flat Earth approximation may not be ideal and the Earth's curvature should be taken into account. In such a case, the Earth could be modeled using tesseroids, also called spherical prisms, instead of the conventional rectangular prisms. The GGT due to a tesseroid is calculated using numerical integration methods, such as the Gauss-Legendre Quadrature (GLQ), as already proposed by Asgharzadeh et al. (2007) and Wild-Pfeiffer (2008). We present a computer program for the direct computation of the GGT caused by a tesseroid using the GLQ. The accuracy of this implementation was evaluated by comparing its results with the result of analytical formulas for the special case of a spherical cap with computation point located at one of the poles. The GGT due to the topographic masses of the Parana basin (SE Brazil) was estimated at 260 km altitude in an attempt to quantify this effect on the GOCE gravity data. The digital elevation model ETOPO1 (Amante and Eakins, 2009) between 40º W and 65º W and 10º S and 35º S, which includes the Paraná Basin, was used to generate a tesseroid model of the topography with grid spacing of 10' x 10' and a constant density of 2670 kg/m3. The largest amplitude observed was on the second vertical derivative component (-0.05 to 1.20 Eötvos) in regions of rough topography, such as that along the eastern Brazilian continental margins. These results indicate that the GGT due to topographic masses may have amplitudes of the same order of magnitude as the GGT due to density anomalies within the crust and mantle.
This document is a master's thesis written in Chinese that investigates the existence and uniqueness of solutions to stochastic differential equations (SDEs) with Lévy noise and non-Lipschitz coefficients. It introduces Lévy processes and their properties, including the Lévy-Itô decomposition. It defines stochastic integration with respect to compensated Poisson processes and provides Itô's formula for Lévy diffusions. The thesis proves that if weak existence and pathwise uniqueness hold for an SDE with Lévy noise, then it has a unique strong solution. It establishes conditions on the coefficients that ensure infinite lifetime and pathwise uniqueness of the solution.
This document discusses relations, functions, and matrices. It defines binary relations as subsets of the Cartesian product of a set with itself that satisfy a given relationship. It discusses properties of relations such as reflexive, symmetric, transitive, and antisymmetric relations. It also defines partial orderings, equivalence relations, and partitioning sets into equivalence classes. Functions are introduced as a special type of binary relation with unique outputs. Matrices are defined as a way to represent linear functions.
Solvability of Matrix Riccati Inequality Talk SlidesKevin Kissi
15min presentation slides. It goes beyond Beamer latex to showcase the best use of color and design in a Mathematical talk slide.
The paper itself is archived at arxiv.org with pdf at: https://arxiv.org/pdf/1505.04861.pdf
This document discusses the breakdown of linearization in the Vlasov equation due to particle trapping in an oscillating electric field. It shows that the linear Landau damping solution is only valid when the perturbation amplitude is small and for times less than the trapping time scale τtr. Beyond τtr, particles become trapped in oscillation in the electric potential well, violating the assumption of unperturbed trajectories used in the linearized solution. Nonlinear effects then dominate and nonlinear methods are required.
This document is a calculus supplement to accompany a microeconomics textbook. It introduces the concept of partial derivatives and shows how they can be used to analyze economic concepts from the textbook like demand and supply functions, substitutes and complements, and elasticities. Partial derivatives allow the slope of demand and supply functions to be determined with respect to different variables. They also allow elasticities to be defined and calculated using calculus, providing an equivalent but alternative method to the algebraic approach in the textbook. The supplement aims to illustrate the connections between calculus and microeconomic concepts.
- Prophecy Resource Corp. released unaudited financial statements for the quarter ended December 31, 2006. The statements have not been reviewed by an auditor.
- During the quarter, the company reported a net loss of $60,207 and ended with $135,119 in cash. The loss was primarily due to $55,304 in stock-based compensation for management fees.
- Cash flows included $4,822 used for deferred exploration costs and $19,000 used for deferred finance fees, while $4,833 was received from related parties.
Charlotte Sale on her experiences on 'Britain's Next Big Thing', from the Hid...Hidden Art
Charlotte Sale on her experiences taking part in the TV programme 'Britain's Next Big Thing' and selling her products to Liberty, taken from the Hidden Art Forum 2011: Making Sales
Mongolia faces an increasing power deficit as its economy grows at over 10% annually. It currently relies on costly power imports from Russia but needs new domestic generation. Prophecy Coal proposes the 600 MW Chandgana Power Plant, fueled by low-sulfur coal from its nearby mine. The plant aims to reduce emissions and costs while increasing energy independence and potential exports. Prophecy has operated in Mongolia for over a year and requests government support to quickly approve permits to begin construction in 2013 and help meet rising demand.
Prophecy Resource Corp. is focused on acquiring base and precious metal exploration properties in British Columbia. In 2008, the company conducted drilling programs at its Okeover copper-molybdenum property, located north of Powell River, BC. Drilling expanded the known mineralized area at the North Lake Zone and intersected intercepts grading up to 0.41% copper and 0.001% molybdenum over 12 meters. Prophecy has now earned a 60% interest in the Okeover property from Eastfield Resources Ltd. by spending over $1 million on exploration as required.
Clare Bristow, Graduate Trainee 2010-11, History Faculty Library, University of Oxford.
Presentation given at the Gradaute Trainee Project Showcase, 13 July 2011.
Systems Evolution, Inc. (SEI) is a business and technology consulting firm that has been providing pragmatic solutions to clients for almost 20 years. SEI focuses on client success through project planning and execution, technology deployment, business process optimization, and enterprise information management. The company's reputation is built on commitment to clients and reliable results.
The document describes the Guitar Musicaid Sliderule, a handheld instructional tool for guitar players. It contains (1) a guitar fingerboard card and slider that reveals note names, fret positions, and fingering for major and minor scales and chords. (2) Users can see the notes, learn finger placement by using the transparent openings on the slider. (3) It helps guitar players of all ages and abilities learn notes and scales easily and is compact for carrying anywhere.
The audited financial statements summarize Prophecy Resource Corp.'s financial position as of September 30, 2009 and 2008, and its results of operations and cash flows for the years then ended. The auditors expressed an opinion that the financial statements present fairly the financial position of the company in accordance with Canadian generally accepted accounting principles. The company is a mineral exploration company that has not yet determined if its properties contain economically recoverable mineral reserves.
El documento presenta una tabla con los nombres, notas y repetición de nombres y notas de 6 alumnos. La nota más alta fue de 19.5, obtenida por Carlos Aguilar, y la más baja fue 8.01, obtenida por Maria Gomez.
- The document provides an overview and analysis of Prophecy Resource Corp's financial condition and operations for the fiscal year ending September 30, 2007. It summarizes the company's exploration activities on its Okeover copper-molybdenum property in British Columbia.
- Drilling in 2007 expanded the boundaries of the property's North Lake Zone, with several holes intersecting significant copper and molybdenum mineralization.
- An independent report estimates the North Lake Zone contains 86.8 million tonnes of inferred resources at average grades of 0.31% copper and 0.014% molybdenum. The zone remains open for further expansion.
Margaret Mead was an influential 20th century anthropologist who studied how culture and environment shape human behavior and development. Her most famous work, Coming of Age in Samoa, analyzed the transition to adulthood in Samoan society and found that Samoan adolescents experienced less stress and conflict compared to American youth. However, Mead's work has also faced criticism for potentially misrepresenting Samoan culture. Overall, Mead was an early proponent of the view that gender roles and social norms are culturally determined rather than biologically fixed.
- The document provides a management discussion and analysis for Prophecy Resource Corp for the three and six months ended June 30, 2010.
- Key events included acquiring additional mining properties through a business combination, raising $4.3 million through share issuances, and reporting a net loss of $1.6 million.
- The company operates mining projects in Canada, Mongolia, and Nevada and is focused on developing its coal, nickel, and platinum group metal properties.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
- Prophecy Resource Corp. released unaudited financial statements for the quarter ended December 31, 2006. The statements have not been reviewed by an auditor.
- During the quarter, the company reported a net loss of $60,207 and ended with $135,119 in cash. The loss was primarily due to $55,304 in stock-based compensation for management fees.
- Cash flows included $4,822 used for deferred exploration costs and $19,000 used for deferred finance fees, while $4,833 was received from related parties.
Charlotte Sale on her experiences on 'Britain's Next Big Thing', from the Hid...Hidden Art
Charlotte Sale on her experiences taking part in the TV programme 'Britain's Next Big Thing' and selling her products to Liberty, taken from the Hidden Art Forum 2011: Making Sales
Mongolia faces an increasing power deficit as its economy grows at over 10% annually. It currently relies on costly power imports from Russia but needs new domestic generation. Prophecy Coal proposes the 600 MW Chandgana Power Plant, fueled by low-sulfur coal from its nearby mine. The plant aims to reduce emissions and costs while increasing energy independence and potential exports. Prophecy has operated in Mongolia for over a year and requests government support to quickly approve permits to begin construction in 2013 and help meet rising demand.
Prophecy Resource Corp. is focused on acquiring base and precious metal exploration properties in British Columbia. In 2008, the company conducted drilling programs at its Okeover copper-molybdenum property, located north of Powell River, BC. Drilling expanded the known mineralized area at the North Lake Zone and intersected intercepts grading up to 0.41% copper and 0.001% molybdenum over 12 meters. Prophecy has now earned a 60% interest in the Okeover property from Eastfield Resources Ltd. by spending over $1 million on exploration as required.
Clare Bristow, Graduate Trainee 2010-11, History Faculty Library, University of Oxford.
Presentation given at the Gradaute Trainee Project Showcase, 13 July 2011.
Systems Evolution, Inc. (SEI) is a business and technology consulting firm that has been providing pragmatic solutions to clients for almost 20 years. SEI focuses on client success through project planning and execution, technology deployment, business process optimization, and enterprise information management. The company's reputation is built on commitment to clients and reliable results.
The document describes the Guitar Musicaid Sliderule, a handheld instructional tool for guitar players. It contains (1) a guitar fingerboard card and slider that reveals note names, fret positions, and fingering for major and minor scales and chords. (2) Users can see the notes, learn finger placement by using the transparent openings on the slider. (3) It helps guitar players of all ages and abilities learn notes and scales easily and is compact for carrying anywhere.
The audited financial statements summarize Prophecy Resource Corp.'s financial position as of September 30, 2009 and 2008, and its results of operations and cash flows for the years then ended. The auditors expressed an opinion that the financial statements present fairly the financial position of the company in accordance with Canadian generally accepted accounting principles. The company is a mineral exploration company that has not yet determined if its properties contain economically recoverable mineral reserves.
El documento presenta una tabla con los nombres, notas y repetición de nombres y notas de 6 alumnos. La nota más alta fue de 19.5, obtenida por Carlos Aguilar, y la más baja fue 8.01, obtenida por Maria Gomez.
- The document provides an overview and analysis of Prophecy Resource Corp's financial condition and operations for the fiscal year ending September 30, 2007. It summarizes the company's exploration activities on its Okeover copper-molybdenum property in British Columbia.
- Drilling in 2007 expanded the boundaries of the property's North Lake Zone, with several holes intersecting significant copper and molybdenum mineralization.
- An independent report estimates the North Lake Zone contains 86.8 million tonnes of inferred resources at average grades of 0.31% copper and 0.014% molybdenum. The zone remains open for further expansion.
Margaret Mead was an influential 20th century anthropologist who studied how culture and environment shape human behavior and development. Her most famous work, Coming of Age in Samoa, analyzed the transition to adulthood in Samoan society and found that Samoan adolescents experienced less stress and conflict compared to American youth. However, Mead's work has also faced criticism for potentially misrepresenting Samoan culture. Overall, Mead was an early proponent of the view that gender roles and social norms are culturally determined rather than biologically fixed.
- The document provides a management discussion and analysis for Prophecy Resource Corp for the three and six months ended June 30, 2010.
- Key events included acquiring additional mining properties through a business combination, raising $4.3 million through share issuances, and reporting a net loss of $1.6 million.
- The company operates mining projects in Canada, Mongolia, and Nevada and is focused on developing its coal, nickel, and platinum group metal properties.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.
IGCSE Biology Chapter 14- Reproduction in Plants.pdf
Charla Santiago Numerico
1. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
A priori and a posteriori error analyses of a
two-fold saddle point approach for a nonlinear
Stokes-Darcy coupled problem
´
G ABRIEL N. G ATICA , R ICARDO OYARZ UA ,
F RANCISCO -J AVIER S AYAS .
WONAPDE 2010
´
U NIVERSIDAD DE C ONCEPCI ON – C HILE .
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
2. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Contents
1 T HE COUPLED PROBLEM
2 T HE CONTINUOUS FORMULATION
3 T HE GALERKIN FORMULATION
4 A POSTERIORI ERROR ESTIMATOR
5 N UMERICAL EXAMPLES
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
3. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Geometry of the problem
ν
ΓS
ΩS
t
Σ
ν
ΩD
ΓD ν
Incompressible viscous fluid in ΩS Porous medium in ΩD
(flowing back and forth across Σ) (saturated with the same fluid)
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
4. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Let fS ∈ L2 (ΩS ) and fD ∈ L2 (ΩS ).
0
Coupled problem: Find velocities (uS , uD ) and pressures (pS , pD )
σ S = − pS I + ν uS in ΩS
− div σ S = fS in ΩS
Stokes equations
div uS = 0 in ΩS
uS = 0 on ΓS
uD = − κ (·, | pD |) pD in ΩD
Darcy equations div uD = fD in ΩD
uD · n = 0 on ΓD
uS · n = uD · n on Σ
Coupling terms ν
σ S n + pD n + (uS · t)t = 0 on Σ
κ
ν > 0: fluid viscosity, κ: friction constant
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
5. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Assumption on κ
There exist constants k0 , k1 > 0, such that for all (x, ρ) ∈ ΩD × R+ :
k0 ≤ κ(x, ρ) ≤ k1 ,
∂
k0 ≤ κ(x, ρ) + ρ κ(x, ρ) ≤ k1 , and
∂ρ
| x κ(x, ρ)| ≤ k1 .
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
6. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
We note that
1
div uS = 0 ∈ ΩS ⇒ pS = − trσ S
2
Rewriting the Stokes equations
pS = − 1 trσ S
2 in ΩS
ν −1
σd
S = uS in ΩS
− div σ S = fS in ΩS
uS = 0 on ΓS
where
1
σ d := σ S − tr σ S I.
S
2
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
7. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Additional unknowns
ϕ := −uS ∈ H1/2 (Σ), λ := pD ∈ H 1/2 (Σ)
tD := pD in ΩD
Rewriting the Darcy equations
tD = pD in ΩD
uD = − κ (·, |tD |)tD in ΩD
div uD = fD in ΩD
uD · n = 0 on ΓD
Rewriting the coupling terms
ϕ · n + uD · n = 0 on Σ
σ S n + λn − νκ−1 (ϕ · t)t = 0 on Σ
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
8. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Additional unknowns
ϕ := −uS ∈ H1/2 (Σ), λ := pD ∈ H 1/2 (Σ)
tD := pD in ΩD
Rewriting the Darcy equations
tD = pD in ΩD
uD = − κ (·, |tD |)tD in ΩD
div uD = fD in ΩD
uD · n = 0 on ΓD
Rewriting the coupling terms
ϕ · n + uD · n = 0 on Σ
σ S n + λn − νκ−1 (ϕ · t)t = 0 on Σ
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
9. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Additional unknowns
ϕ := −uS ∈ H1/2 (Σ), λ := pD ∈ H 1/2 (Σ)
tD := pD in ΩD
Rewriting the Darcy equations
tD = pD in ΩD
uD = − κ (·, |tD |)tD in ΩD
div uD = fD in ΩD
uD · n = 0 on ΓD
Rewriting the coupling terms
ϕ · n + uD · n = 0 on Σ
σ S n + λn − νκ−1 (ϕ · t)t = 0 on Σ
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
10. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Spaces
L2 (Ω ) := [L2 (Ω )]2 , ∈ {S, D}
H1/2 (Σ) := [H 1/2 (Σ)]2
H(div ; ΩS ) := τ : ΩS → R2×2 : at τ ∈ H(div ; ΩS ), ∀a ∈ R2
HΓD (div ; ΩD ) := {v ∈ H(div , ΩD ) : v · n = 0 on ΓD }
Unknowns
(σ S , tD ) ∈ H(div ; ΩS ) × L2 (ΩD )
(uS , uD , ϕ) ∈ L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ)
(pD , λ) ∈ L2 (ΩD ) × H 1/2 (Σ)
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
11. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Spaces
L2 (Ω ) := [L2 (Ω )]2 , ∈ {S, D}
H1/2 (Σ) := [H 1/2 (Σ)]2
H(div ; ΩS ) := τ : ΩS → R2×2 : at τ ∈ H(div ; ΩS ), ∀a ∈ R2
HΓD (div ; ΩD ) := {v ∈ H(div , ΩD ) : v · n = 0 on ΓD }
Unknowns
(σ S , tD ) ∈ H(div ; ΩS ) × L2 (ΩD )
(uS , uD , ϕ) ∈ L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ)
(pD , λ) ∈ L2 (ΩD ) × H 1/2 (Σ)
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
12. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Variational equations
ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ
S S Σ = 0 ∀ τ S ∈ H(div ; ΩS )
(div σ S , vS )S = −(fS , vS )S ∀ vS ∈ L2 (ΩS )
(κ (·, |tD |)tD , sD )D + (uD , sD )D = 0 ∀ sD ∈ L2 (ΩD )
(tD , vD )D + (div vD , pD )D + vD · n, λ Σ = 0 ∀ vD ∈ H(div; ΩD )
(div uD , qD )D = (qD , fD )D ∀ qD ∈ L2 (ΩD )
ϕ · n, ξ Σ + uD · n, ξ Σ = 0 ∀ ξ ∈ H 1/2 (Σ)
ν
σ S n, ψ Σ + ψ · n, λ Σ − ψ · t, ϕ · t Σ = 0 ∀ ψ ∈ H1/2 (Σ)
κ
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
13. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Decomposition of σ S
σ S + c I with the new unknowns σ S ∈ H0 (div ; ΩS ) and c ∈ R,
where
H0 (div ; ΩS ) := {τ S ∈ H(div ; ΩS ) : tr (τ S ) = 0}
ΩS
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
14. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ
S S Σ = 0 ∀ τ S ∈ H(div ; ΩS )
ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ
S S Σ = 0 ∀ τ S ∈ H0 (div ; ΩS )
d ϕ · n, 1 Σ = 0 ∀d ∈ R
ν
σ S n, ψ Σ + ψ · n, λ Σ − ϕ · t, ψ · t Σ = 0 ∀ψ in H1/2 (Σ)
κ
ν
σ S n, ψ Σ + ψ·n, λ Σ− ϕ·t, ψ·t Σ + c ψ·n, 1 Σ = 0 ∀ ψ ∈ H1/2 (Σ)
κ
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
15. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ
S S Σ = 0 ∀ τ S ∈ H(div ; ΩS )
ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ
S S Σ = 0 ∀ τ S ∈ H0 (div ; ΩS )
d ϕ · n, 1 Σ = 0 ∀d ∈ R
ν
σ S n, ψ Σ + ψ · n, λ Σ − ϕ · t, ψ · t Σ = 0 ∀ψ in H1/2 (Σ)
κ
ν
σ S n, ψ Σ + ψ·n, λ Σ− ϕ·t, ψ·t Σ + c ψ·n, 1 Σ = 0 ∀ ψ ∈ H1/2 (Σ)
κ
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
16. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Global spaces
X := H(div ; ΩS ) × L2 (ΩD )
M := L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ)
Q := L2 (ΩD ) × H 1/2 (Σ) × R
0
Global unknowns
t := (σ S , tD ) ∈ X
u := (uS , uD , ϕ) ∈ M
p := (pD , λ, c) ∈ Q
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
17. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Global spaces
X := H(div ; ΩS ) × L2 (ΩD )
M := L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ)
Q := L2 (ΩD ) × H 1/2 (Σ) × R
0
Global unknowns
t := (σ S , tD ) ∈ X
u := (uS , uD , ϕ) ∈ M
p := (pD , λ, c) ∈ Q
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
18. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Continuous formulation
Find (t, u, p) := ((σ S , tD ), (uS , uD , ϕ), (pD , λ, c)) ∈ X × M × Q such
that,
[A(t), s] + [B1 (s), u] = [F, s], ∀s ∈ X
[B1 (t), v] − [S(u), v] + [B(v), p] = [G1 , v] ∀v ∈ M
[B(u), q] = [G, q] ∀q ∈ Q
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
19. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Operators and functionals
[A(t), s] := ν −1 (σ d , τ d )S + (κ(·, |tD |)tD , sD )D
S S
[B1 (s), v] := (div τ S , vS )S + (vD , sD )D + τ S n, ψ Σ
[B(v), q] := (div vD , qD )D + vD · n, ξ Σ + ψ · n, ξ Σ + d n, ψ Σ
[S(u), v] := νκ −1
ψ · t, ϕ · t Σ
[F, s] := 0, [G1 , v] := (fS , vS )S
[G, q] := (fD , qD )D
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
20. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Equivalent augmented formulation
Find (t, u, p) := ((σ S , tD ), (uS , uD , ϕ), (pD , λ, c)) ∈ X × M × Q such
that,
˜
[A(t), s] + [B1 (s), u] ˜
= [F, s] ∀s ∈ X
[B1 (t), v] − [S(u), v] + [B(v), p] = [G1 , v] ∀v ∈ M
[B(u), q] = [G, q] ∀q ∈ Q
˜
[A(t), s] := [A(t), s] + (div σ S , div τ S )S
= ν −1 (σ d , τ d )S + (div σ S , div τ S )S + (κ(·, |tD |)tD , sD )D
S S
˜
[F, s] := −(fS , div τ S )
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
21. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Lemma: Inf-sup condition for B
There exists β > 0 such that
[B(v), q]
sup ≥ β q Q ∀ q ∈ Q.
v∈M v M
v=0
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
22. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
kernel(B)
˜
M := v := (vS , vD , ψ) ∈ M : n, ψ = 0, vD · n = −ψ · n on Σ
Σ
and div vD = 0 in ΩD .
Lemma: Inf-sup condition for B1
˜
Let M := kernel(B), that is
˜
M := v ∈ M : [B(v), q] = 0 ∀ q ∈ Q .
Then, there exists β1 > 0 such that
[B1 (s), v] ˜
sup ≥ β1 v M ∀ v ∈ M.
s∈X s X
s=0
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
23. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
kernel(B)
˜
M := v := (vS , vD , ψ) ∈ M : n, ψ = 0, vD · n = −ψ · n on Σ
Σ
and div vD = 0 in ΩD .
Lemma: Inf-sup condition for B1
˜
Let M := kernel(B), that is
˜
M := v ∈ M : [B(v), q] = 0 ∀ q ∈ Q .
Then, there exists β1 > 0 such that
[B1 (s), v] ˜
sup ≥ β1 v M ∀ v ∈ M.
s∈X s X
s=0
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
24. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Lemma
˜
The nonlinear operator A : X → X is strongly monotone and
Lipschitz continuous, that is, there exist α, γ > 0 such that
˜ ˜
[A(t) − A(r), t − r] ≥ α t − r 2
X
and
˜ ˜
[A(t) − A(r), s] ≤ γ t − r X s X,
for all t, r, s ∈ X.
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
25. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Theorem
For each (F, G1 , G) ∈ X × M × Q there exists a unique
(t, u, p) ∈ X × M × Q such that
[A(t), s] + [B1 (s), u] = [F, s], ∀ s ∈ X,
[B1 (t), v] − [S(u), v] + [B(v), p] = [G1 , v] ∀ v ∈ M,
[B(u), q] = [G, q] ∀ q ∈ Q,
Moreover, there exists a constant C > 0, independent of the solution,
such that
(t, u, p) X×M×Q ≤ C{ F X + G1 M + G Q }.
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
26. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Discrete spaces ( ∈ {S, D})
Hh (Ω ) ⊆ H(div ; Ω ) , Lh (Ω ) ⊆ L2 (Ω ) , Λh (Σ) ⊆ H 1/2 (Σ)
Lh (Ω ) := [Lh (Ω )]2 , ΛS (Σ) := [ΛS (Σ)]2
h h
Hh (ΩS ) := { τ : ΩS → R2×2 : ct τ ∈ Hh (ΩS ) ∀ c ∈ R2 },
Hh,ΓD := v ∈ Hh (ΩD ) : v · n = 0 on ΓD
Hh,0 (ΩS ) := Hh (ΩS ) ∩ H0 (div ; ΩS ), Lh,0 (ΩD ) := Lh (ΩD ) ∩ L2 (ΩD ) .
0
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
27. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Global discrete spaces
Xh := Hh,0 (ΩS ) × Lh (ΩD )
Mh := Lh (ΩD ) × Hh,ΓD (ΩD ) × ΛS (Σ)
h
Qh := Lh,0 (ΩD ) × ΛD (Σ) × R
h
Global discrete unknowns
th := (σ S,h , tD,h ) ∈ Xh
uh := (uS,h , uD,h , ϕh ) ∈ Mh
ph := (pD,h , λh , ch ) ∈ Qh
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
28. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Global discrete spaces
Xh := Hh,0 (ΩS ) × Lh (ΩD )
Mh := Lh (ΩD ) × Hh,ΓD (ΩD ) × ΛS (Σ)
h
Qh := Lh,0 (ΩD ) × ΛD (Σ) × R
h
Global discrete unknowns
th := (σ S,h , tD,h ) ∈ Xh
uh := (uS,h , uD,h , ϕh ) ∈ Mh
ph := (pD,h , λh , ch ) ∈ Qh
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
29. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Galerkin scheme
[A(th ), s] + [B1 (s), uh ] = [F, s] ∀ s ∈ Xh ,
[B1 (th ), v] − [S(uh ), v] + [B(v), ph ] = [G1 , v] ∀ v ∈ Mh ,
[B(uh ), q] = [G, q] ∀ q ∈ Qh ;
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
30. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Equivalent augmented galerkin scheme
˜
[A(th ), s] + [B1 (s), uh ] ˜
= [F, s] ∀ s ∈ Xh ,
[B1 (th ), v] − [S(uh ), v] + [B(v), ph ] = [G1 , v] ∀ v ∈ Mh ,
[B(uh ), q] = [G, q] ∀ q ∈ Qh ;
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
31. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Discrete Hypothesis
(H.0) [P0 (ΩS )]2 ⊆ Hh (ΩS ) and P0 (ΩD ) ⊆ Lh (ΩD ).
(H.1) There exist βD > 0, independent of h and there exists
ψ 0 ∈ H1/2 (Σ), such that
qh div vh + vh · n, ξh Σ
ΩD
sup ≥ βD qh 0,ΩD + ξh 1/2,Σ
vh ∈ Hh,ΓD (ΩD )0 vh div ,ΩD
∀ (qh , ξh ) ∈ Lh,0 (ΩD ) × ΛD (Σ),
h
ψ 0 ∈ ΛS (Σ) ∀ h and
h ψ 0 · n, 1 Σ = 0.
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
32. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Discrete Hypothesis
(H.2) div Hh (ΩD ) ⊆ Lh (ΩD ).
˜
(H.3) Hh (ΩD ) ⊆ Lh (ΩD ), and there exists βS , independent of h, such
that
vh div τ h + τ h · n, ψh Σ
ΩS
sup ≥ βS vh 0,ΩS + ψh 1/2,Σ
τ h ∈Hh (ΩS )0 τh div ,ΩS
∀ (vh , ψh ) ∈ Lh (ΩS ) × ΛS (Σ), where
h
˜
Hh (ΩD ) := vh ∈ Hh (ΩD ) : div vh = 0 .
(H.4) div Hh (ΩS ) ⊆ Lh (ΩS ).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
33. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Theorem
Assume that (H.0), (H.1), (H.2), (H.3) and (H.4) hold. Then there
exists a unique (th , uh , ph ) ∈ Xh × Mh × Qh such that
[A(th ), s] + [B1 (s), uh ] = [F, s] ∀ s ∈ Xh ,
[B1 (th ), v] − [S(uh ), v] + [B(v), ph ] = [G1 , v] ∀ v ∈ Mh ,
[B(uh ), q] = [G, q] ∀ q ∈ Qh
˜
Moreover there exist C, C > 0, independent of h, such that
(th , uh , ph ) ≤ C F|Xh Xh + G1 |Mh Mh + G|Qh Qh .
˜
(t−th , u−uh , p−ph ) ≤ C inf t−sh X + inf u−vh M + inf p−ph Q
sh ∈Xh vh ∈Mh vh ∈Qh
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
34. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Particular choice of discrete spaces
Let Th and Th be respective triangulations of the domains ΩS and
S D
ΩD .
S D
Raviart–Thomas space of the lowest order (T ∈ Th ∪ Th )
RT0 (T ) := span (1, 0), (0, 1), (x1 , x2 ) .
Discrete spaces in Ω ( ∈ {S, D})
Hh (Ω ) := vh ∈ H(div ; Ω ) : vh |T ∈ RT0 (T ) ∀ T ∈ Th ,
Lh (Ω ) := qh : Ω → R : qh |T ∈ P0 (T ) ∀ T ∈ Th .
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
35. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Particular choice of discrete spaces
Let Th and Th be respective triangulations of the domains ΩS and
S D
ΩD .
S D
Raviart–Thomas space of the lowest order (T ∈ Th ∪ Th )
RT0 (T ) := span (1, 0), (0, 1), (x1 , x2 ) .
Discrete spaces in Ω ( ∈ {S, D})
Hh (Ω ) := vh ∈ H(div ; Ω ) : vh |T ∈ RT0 (T ) ∀ T ∈ Th ,
Lh (Ω ) := qh : Ω → R : qh |T ∈ P0 (T ) ∀ T ∈ Th .
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
36. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Particular choice of discrete spaces
Discrete spaces on the interface (ΛS (Σ) = ΛD (Σ) = Λh (Σ) )
h h
Let us assume that the number of edges of Σh is an even number
and there exists c > 0, independent of h, such that
max |e1 |, |e2 | ≤ c min |e1 |, |e2 | ,
for each pair e1 , e2 ∈ Σh such that e1 ∪ e2 ∈ Σ2h . Then, we let Σ2h
be the partition of Σ arising by joining pairs of adjacent elements, and
define
Λh (Σ) := P1 (Σ2h ) ∩ C(Σ) .
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the Stokes-Darcy
coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de Concepcion, (2009).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
37. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
A posteriori error indicator for ΩS
Θ2 :=
S,T fS + div σ S,h 2
0,T + h2 σ d
T S,h
2
0,T + h2 rot σ d
T S,h
2
0,T
ν 2
+ he (σ S,h + ch I)n + λh n − (ϕ · t)t
κ h 0,e
e∈Eh (T )∩Eh (Σ)
2
+ he ν −1 σ d t +
S,h ϕh t + he ϕh + uS,h 2
0,e
0,e
+ he [σ d t]
S,h
2
0,e + he [uS,h ] 2
0,e
e∈E(T )∩(Eh (ΩS )∪Eh (ΓS ))
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
38. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
A posteriori error indicator for ΩD
Θ2 :=
D,T fD − div uD,h 2
0,T + h2 rot (tD,h )
T
2
0,T
+ h2 tD,h
T
2
0,T + κ(·, |tD,h |)tD,h + uD,h 0,T
2
+ he [pD,h ] 2
0,e + he [tD,h · t] 0,e
e∈E(T )∩(Eh (ΩD )∪Eh (ΓD ))
2
dλh 2
+ he tD,h · t − + he ϕh · n + uD,h · n 0,e
dt 0,e
e∈E(T )∩Eh (Σ)
+ he pD,h − λh 2
0,e
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
39. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Theorem
˜
There exist Crel , Cef f > 0, independent of h and h such that
Cef f Θ ≤ σ − σ h X + u − uh M + p − ph Q ≤ Crel Θ,
where 1/2
Θ = Θ2 +
S,T Θ2
D,T .
S D
T ∈Th T ∈Th
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
40. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Notations
e(tD ) := tD − tD,h 0,ΩD e(uS ) := uS − uS,h 0,ΩS ,
e(pD ) := pD − pD,h 0,ΩD , e(uD ) := uD − uD,h div ,ΩD
e(σ S ) := σ S − σ S,h div ,ΩS , e(λ) := λ − λh 1/2,Σ
e(ϕ) := ϕ − ϕh 1/2,Σ
1/2
eT := e(tD )2 + e(uS )2 + e(pD )2 + e(uD )2 + e(σ S )2 + e(λ)2 + e(ϕ)2
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
41. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Example 1. µ = 1, κ = 1.
ΩS := (−1, 1) × (0, 1) ,
ΩD := (−1, 1) × (−1, 0) ,
uS (x, y) := curl (x2 − 1)2 (y − 1)2 ,
pS (x, y) := x3 + y 3 ,
pD (x, y) := sin(πx)3 (y + 1)2 ,
1
κ(·, s) := 2 + .
1+s
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
42. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Errors and rates of convergence.
N e(tD ) r(tD ) e(uS ) r(uS ) e(pD ) r(pD ) e(uD ) r(uD ) e(σ S ) r(σ S )
172 2.072 — 0.449 — 0.718 — 4.714 — 2.837 —
644 0.988 1.123 0.233 0.997 0.180 2.099 2.176 1.171 1.287 1.197
2500 0.535 0.904 0.115 1.045 0.060 1.622 1.123 0.976 0.619 1.099
9860 0.250 1.106 0.057 1.019 0.026 1.193 0.522 1.117 0.302 1.027
39172 0.124 1.019 0.028 1.007 0.013 1.032 0.258 1.021 0.151 1.008
156164 0.062 1.004 0.014 1.003 0.006 1.008 0.129 1.005 0.075 1.003
N ˜
h e(λ) r(λ) e(ϕ) r(ϕ)
172 0.998 1.174 — 2.859 —
644 0.499 0.892 0.417 1.187 1.332
2500 0.250 0.490 0.884 0.534 1.178
9860 0.125 0.217 1.187 0.258 1.061
39172 0.062 0.105 1.051 0.128 1.018
156164 0.031 0.052 1.014 0.064 1.006
N eT r(eT ) Θ r(Θ) eT /Θ
172 6.6958 — 4.2323 — 1.5821
644 3.1075 1.1629 2.0629 1.0887 1.5064
2500 1.5690 1.0077 0.9789 1.0992 1.6028
9860 0.7374 1.1005 0.4929 1.0001 1.4961
39172 0.3647 1.0209 0.2461 1.0071 1.4819
156164 0.1820 1.0054 0.1233 0.9990 1.4754
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
43. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Example 2, µ = 1, κ = 1.
ΩS := (−1, 1) × (0, 1)[0, 1] × [0.5, 1] ,
ΩD := (−1, 1) × (−1, 0) [0, 1] × [−1, −0.5] ,
x2 (x2 − 1)2 (y − 1)2 (y − 0.5)2
uS (x, y) = curl ,
(x2 + (y − 0.5)2 + 0.01)2
pS (x, y) = sin(2πx)3 (y + 1)2 (y + 0.5)2 ,
pD (x, y) = sin(2πx)3 (y + 1)2 (y + 0.5)2 ,
1
κ(·, s) := 2 + .
1+s
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
44. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Uniform refinement.
N eT r(eT ) Θ eT /Θ
110 39.0943 — 38.8240 1.0070
396 53.6275 — 54.1512 0.9903
1508 65.9962 — 66.4284 0.9935
5892 45.7855 0.5366 46.0250 0.9948
23300 24.1982 0.9276 24.2902 0.9962
92676 12.9528 0.9053 12.9270 1.0020
Adaptive refinement.
N eT r(eT ) Θ e/Θ
110 39.0943 — 38.8240 1.0070
289 54.9264 — 55.3642 0.9921
479 66.9987 — 67.5925 0.9912
657 52.4883 1.5449 53.0637 0.9892
1315 35.2800 1.1450 35.9303 0.9819
3759 21.5520 0.9385 21.8753 0.9852
4017 20.4178 1.6288 20.7288 0.9850
7875 15.6882 0.7829 15.9045 0.9864
10191 13.5102 1.1595 13.7352 0.9836
16558 10.4623 1.0535 10.5737 0.9895
28745 7.9688 0.9871 8.0054 0.9954
48715 6.1675 0.9715 6.1166 1.0083
70713 5.1460 0.9718 5.0318 1.0227
109264 4.2753 0.8520 4.0772 1.0486
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
45. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Errors vs degrees of freedom.
100
adaptative ♦
♦ + uniform +
♦ + ♦
+
+
♦ ♦
+
♦
♦
e ♦
♦ +
10 ♦
♦
♦
♦
♦
100 1000 10000 100000
N
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
46. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
Grids with 110, 1315, 4017 and 28745 degrees of freedom.
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
47. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
48. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
49. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
50. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
51. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
52. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
53. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
54. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem
55. T HE COUPLED PROBLEM
T HE CONTINUOUS FORMULATION
T HE GALERKIN FORMULATION
A POSTERIORI ERROR ESTIMATOR
N UMERICAL EXAMPLES
ˇ
I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical
Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).
F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point
problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).
´
G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of
fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the
Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.
´
G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the
Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de
Concepcion, (2009).
G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and
nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).
G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische
Matematik, vol. 109,2,pp 209-231, (2008).
W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on
Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).
G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas
´ Stokes-Darcy coupled problem