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Anisotropic mesh refinement for singularly perturbed reaction diffusion problemsApel, Th., Lube, G. 30 October 1998 (has links) (PDF)
The paper is concerned with the finite element resolution of layers appearing
in singularly perturbed problems. A special anisotropic grid of Shishkin type
is constructed for reaction diffusion problems. Estimates of the finite element
error in the energy norm are derived for two methods, namely the standard
Galerkin method and a stabilized Galerkin method. The estimates are uniformly
valid with respect to the (small) diffusion parameter. One ingredient is a
pointwise description of derivatives of the continuous solution. A numerical
example supports the result.
Another key ingredient for the error analysis is a refined estimate for
(higher) derivatives of the interpolation error. The assumptions on admissible
anisotropic finite elements are formulated in terms of geometrical conditions
for triangles and tetrahedra. The application of these estimates is not
restricted to the special problem considered in this paper.
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Coupling Methods for Interior Penalty Discontinuous Galerkin Finite Element Methods and Boundary Element MethodsOf, Günther, Rodin, Gregory J., Steinbach, Olaf, Taus, Matthias 19 October 2012 (has links) (PDF)
This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.
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Anisotropic mesh refinement in stabilized Galerkin methodsApel, Thomas, Lube, Gert 30 October 1998 (has links) (PDF)
The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.
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ITERATIVE SOLVERS FOR DISCONTINUOUS GALERKIN FINITE ELEMENT METHODSSINGH, ONKAR DEEP 06 October 2004 (has links)
No description available.
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Anisotropic mesh refinement in stabilized Galerkin methodsApel, Thomas, Lube, Gert 30 October 1998 (has links)
The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.
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Anisotropic mesh refinement for singularly perturbed reaction diffusion problemsApel, Th., Lube, G. 30 October 1998 (has links)
The paper is concerned with the finite element resolution of layers appearing
in singularly perturbed problems. A special anisotropic grid of Shishkin type
is constructed for reaction diffusion problems. Estimates of the finite element
error in the energy norm are derived for two methods, namely the standard
Galerkin method and a stabilized Galerkin method. The estimates are uniformly
valid with respect to the (small) diffusion parameter. One ingredient is a
pointwise description of derivatives of the continuous solution. A numerical
example supports the result.
Another key ingredient for the error analysis is a refined estimate for
(higher) derivatives of the interpolation error. The assumptions on admissible
anisotropic finite elements are formulated in terms of geometrical conditions
for triangles and tetrahedra. The application of these estimates is not
restricted to the special problem considered in this paper.
|
7 |
Coupling Methods for Interior Penalty Discontinuous Galerkin Finite Element Methods and Boundary Element MethodsOf, Günther, Rodin, Gregory J., Steinbach, Olaf, Taus, Matthias 19 October 2012 (has links)
This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.:1. Introduction
2. Model Problem and Background
3. New Coupling Methods
4. Stability and Error Analysis
5. Numerical Examples
6. Summary
A. Appendix
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