Error estimates for anisotropic finite elements and applications
The finite element method is one of the most frequently used techniques to approximate the solution of partial differential equations. It consists in approximating the unknown solution by functions which are polynomials on each element of a given partition of the domain, made of triangles or quadril...
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todo:paper_NIS06866_v3_n_p1181_Duran2023-10-03T16:45:46Z Error estimates for anisotropic finite elements and applications Durán, R.G. Anisotropic elements Convection-diffusion Finite elements Mixed methods Stokes equations Anisotropic elements Convection diffusion Finite Element Mixed method Stokes equations Anisotropy Diffusion in liquids Error analysis Finite element method Heat convection Interpolation Partial differential equations Polynomials Estimation The finite element method is one of the most frequently used techniques to approximate the solution of partial differential equations. It consists in approximating the unknown solution by functions which are polynomials on each element of a given partition of the domain, made of triangles or quadrilaterals (or their generalizations to higher dimensions). A fundamental problem is to estimate the error between the exact solution u and its computable finite element approximation. In many situations this error can be bounded in terms of the best approximation of u by functions in the finite element space of piecewise polynomial functions. A natural way to estimate this best approximation is by means of the Lagrange interpolation or other similar procedures. Many works have considered the problem of interpolation error estimates. The classical error analysis for interpolations is based on the so-called regularity assumption, which excludes elements with different sizes in each direction (called anisotropic). The goal of this paper is to present a different approach which has been developed by many authors and can be applied to obtain error estimates for several interpolations under more general hypotheses. An important case in which anisotropic elements arise naturally is in the approximation of convection-diffusion problems which present boundary layers. We present some applications to these problems. Finally we consider the finite element approximation of the Stokes equations and present some results for non-conforming methods. © 2006 European Mathematical Society. Fil:Durán, R.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. CONF info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_NIS06866_v3_n_p1181_Duran |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
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R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Anisotropic elements Convection-diffusion Finite elements Mixed methods Stokes equations Anisotropic elements Convection diffusion Finite Element Mixed method Stokes equations Anisotropy Diffusion in liquids Error analysis Finite element method Heat convection Interpolation Partial differential equations Polynomials Estimation |
| spellingShingle |
Anisotropic elements Convection-diffusion Finite elements Mixed methods Stokes equations Anisotropic elements Convection diffusion Finite Element Mixed method Stokes equations Anisotropy Diffusion in liquids Error analysis Finite element method Heat convection Interpolation Partial differential equations Polynomials Estimation Durán, R.G. Error estimates for anisotropic finite elements and applications |
| topic_facet |
Anisotropic elements Convection-diffusion Finite elements Mixed methods Stokes equations Anisotropic elements Convection diffusion Finite Element Mixed method Stokes equations Anisotropy Diffusion in liquids Error analysis Finite element method Heat convection Interpolation Partial differential equations Polynomials Estimation |
| description |
The finite element method is one of the most frequently used techniques to approximate the solution of partial differential equations. It consists in approximating the unknown solution by functions which are polynomials on each element of a given partition of the domain, made of triangles or quadrilaterals (or their generalizations to higher dimensions). A fundamental problem is to estimate the error between the exact solution u and its computable finite element approximation. In many situations this error can be bounded in terms of the best approximation of u by functions in the finite element space of piecewise polynomial functions. A natural way to estimate this best approximation is by means of the Lagrange interpolation or other similar procedures. Many works have considered the problem of interpolation error estimates. The classical error analysis for interpolations is based on the so-called regularity assumption, which excludes elements with different sizes in each direction (called anisotropic). The goal of this paper is to present a different approach which has been developed by many authors and can be applied to obtain error estimates for several interpolations under more general hypotheses. An important case in which anisotropic elements arise naturally is in the approximation of convection-diffusion problems which present boundary layers. We present some applications to these problems. Finally we consider the finite element approximation of the Stokes equations and present some results for non-conforming methods. © 2006 European Mathematical Society. |
| format |
CONF |
| author |
Durán, R.G. |
| author_facet |
Durán, R.G. |
| author_sort |
Durán, R.G. |
| title |
Error estimates for anisotropic finite elements and applications |
| title_short |
Error estimates for anisotropic finite elements and applications |
| title_full |
Error estimates for anisotropic finite elements and applications |
| title_fullStr |
Error estimates for anisotropic finite elements and applications |
| title_full_unstemmed |
Error estimates for anisotropic finite elements and applications |
| title_sort |
error estimates for anisotropic finite elements and applications |
| url |
http://hdl.handle.net/20.500.12110/paper_NIS06866_v3_n_p1181_Duran |
| work_keys_str_mv |
AT duranrg errorestimatesforanisotropicfiniteelementsandapplications |
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1807316579611836416 |