H P -finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form

Andrea Toselli

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 1, page 91-115
  • ISSN: 0764-583X

Abstract

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We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect to the mesh–size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape–regular meshes and arbitrary partitions into subdomains. Our method can be applied to advective, diffusive, and mixed–type equations, as well, and is well-suited for problems coupling hyperbolic and elliptic equations. We present some two-dimensional numerical results that support our analysis for the case of linear finite elements.

How to cite

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Toselli, Andrea. "${HP}$-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.1 (2003): 91-115. <http://eudml.org/doc/244912>.

@article{Toselli2003,
abstract = {We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect to the mesh–size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape–regular meshes and arbitrary partitions into subdomains. Our method can be applied to advective, diffusive, and mixed–type equations, as well, and is well-suited for problems coupling hyperbolic and elliptic equations. We present some two-dimensional numerical results that support our analysis for the case of linear finite elements.},
author = {Toselli, Andrea},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Advection–diffusion; hyperbolic problems; stabilization; domain decomposition; non-matching grids; discontinuous Galerkin; $hp$-finite elements; Advection-diffusion equation; discontinuous Galerkin method; hp-finite elements; error bounds; numerical results},
language = {eng},
number = {1},
pages = {91-115},
publisher = {EDP-Sciences},
title = {$\{HP\}$-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form},
url = {http://eudml.org/doc/244912},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Toselli, Andrea
TI - ${HP}$-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 1
SP - 91
EP - 115
AB - We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect to the mesh–size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape–regular meshes and arbitrary partitions into subdomains. Our method can be applied to advective, diffusive, and mixed–type equations, as well, and is well-suited for problems coupling hyperbolic and elliptic equations. We present some two-dimensional numerical results that support our analysis for the case of linear finite elements.
LA - eng
KW - Advection–diffusion; hyperbolic problems; stabilization; domain decomposition; non-matching grids; discontinuous Galerkin; $hp$-finite elements; Advection-diffusion equation; discontinuous Galerkin method; hp-finite elements; error bounds; numerical results
UR - http://eudml.org/doc/244912
ER -

References

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