Numerical Modelling and Capturing Singularities

분야Field	Math Colloquium
날짜Date	2019-05-24 ~ 2019-05-24	시간Time	15:50 ~ 18:00
장소Place	Math. Bldg. 404	초청자Host	최민석
연사Speaker	Eun-Jae Park	소속Affiliation	Yonsei Univ.
TOPIC	Numerical Modelling and Capturing Singularities
소개 및 안내사항Content	Part 1. Title: Numerical Modelling and Capturing Singularities Abstract: Mathematical models that involve convective and/or diffusive processes are among the most widespread in all of science and engineering. The purpose of this talk is to present what numerical modelling is about and to introduce finite element methods for the numerical solution of partial differential equations in mechanics and physics. We often encounter singularities from convection diffusion equations and/or the problem domain. We introduce the idea of adaptive mesh refinement to efficiently capture singularities via a posteriori error analysis. Several numerical examples will be presented. Part 2. Title: Staggered Galerkin Method:  A New Computational Paradigm Abstract: Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems. In this talk,  a new computational paradigm for discretizing PDEs is presented via staggered Galerkin method on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. We derive a simple residual-type error estimator.  Numerical results indicate that optimal convergence can be achieved for both the potential  and vector variables, and the singularity can be well-captured by the proposed error estimator.  Then, some applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this approach to high-order polynomial approximations on general meshes.

학회명Field	Numerical Modelling and Capturing Singularities
날짜Date	2019-05-24 ~ 2019-05-24	시간Time	15:50 ~ 18:00
장소Place	Math. Bldg. 404	초청자Host	최민석
소개 및 안내사항Content	Part 1. Title: Numerical Modelling and Capturing Singularities Abstract: Mathematical models that involve convective and/or diffusive processes are among the most widespread in all of science and engineering. The purpose of this talk is to present what numerical modelling is about and to introduce finite element methods for the numerical solution of partial differential equations in mechanics and physics. We often encounter singularities from convection diffusion equations and/or the problem domain. We introduce the idea of adaptive mesh refinement to efficiently capture singularities via a posteriori error analysis. Several numerical examples will be presented. Part 2. Title: Staggered Galerkin Method:  A New Computational Paradigm Abstract: Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems. In this talk,  a new computational paradigm for discretizing PDEs is presented via staggered Galerkin method on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. We derive a simple residual-type error estimator.  Numerical results indicate that optimal convergence can be achieved for both the potential  and vector variables, and the singularity can be well-captured by the proposed error estimator.  Then, some applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this approach to high-order polynomial approximations on general meshes.

성명Field	Numerical Modelling and Capturing Singularities
날짜Date	2019-05-24 ~ 2019-05-24	시간Time	15:50 ~ 18:00
소속Affiliation	Yonsei Univ.	초청자Host	최민석
소개 및 안내사항Content	Part 1. Title: Numerical Modelling and Capturing Singularities Abstract: Mathematical models that involve convective and/or diffusive processes are among the most widespread in all of science and engineering. The purpose of this talk is to present what numerical modelling is about and to introduce finite element methods for the numerical solution of partial differential equations in mechanics and physics. We often encounter singularities from convection diffusion equations and/or the problem domain. We introduce the idea of adaptive mesh refinement to efficiently capture singularities via a posteriori error analysis. Several numerical examples will be presented. Part 2. Title: Staggered Galerkin Method:  A New Computational Paradigm Abstract: Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems. In this talk,  a new computational paradigm for discretizing PDEs is presented via staggered Galerkin method on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. We derive a simple residual-type error estimator.  Numerical results indicate that optimal convergence can be achieved for both the potential  and vector variables, and the singularity can be well-captured by the proposed error estimator.  Then, some applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this approach to high-order polynomial approximations on general meshes.

성명Field	Numerical Modelling and Capturing Singularities
날짜Date	2019-05-24 ~ 2019-05-24	시간Time	15:50 ~ 18:00
호실Host		인원수Affiliation	Eun-Jae Park
사용목적Affiliation	최민석	신청방식Host	Yonsei Univ.
소개 및 안내사항Content	Part 1. Title: Numerical Modelling and Capturing Singularities Abstract: Mathematical models that involve convective and/or diffusive processes are among the most widespread in all of science and engineering. The purpose of this talk is to present what numerical modelling is about and to introduce finite element methods for the numerical solution of partial differential equations in mechanics and physics. We often encounter singularities from convection diffusion equations and/or the problem domain. We introduce the idea of adaptive mesh refinement to efficiently capture singularities via a posteriori error analysis. Several numerical examples will be presented. Part 2. Title: Staggered Galerkin Method:  A New Computational Paradigm Abstract: Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems. In this talk,  a new computational paradigm for discretizing PDEs is presented via staggered Galerkin method on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. We derive a simple residual-type error estimator.  Numerical results indicate that optimal convergence can be achieved for both the potential  and vector variables, and the singularity can be well-captured by the proposed error estimator.  Then, some applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this approach to high-order polynomial approximations on general meshes.