Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem

Papri Majumder

Applications of Mathematics (2021)

  • Volume: 66, Issue: 5, page 673-699
  • ISSN: 0862-7940

Abstract

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We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in ℝ d ( d = 2 , 3 ) . For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete schemes, and derive error estimate of optimal order 𝒪 ( h + Δ t ) in an energy norm. Moreover, the analysis is performed without any assumptions on the speed of propagation of the free boundary and only the realistic regularity u t ∈ ℒ 2 ( 0 , T ; ℒ 2 ( Ω ) ) is assumed. Further, we present some numerical experiments to illustrate the performance of the proposed methods.

How to cite

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Majumder, Papri. "Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem." Applications of Mathematics 66.5 (2021): 673-699. <http://eudml.org/doc/298218>.

@article{Majumder2021,
abstract = {We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in $\mathbb \{R\}^d$$(d=2,3)$. For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete schemes, and derive error estimate of optimal order $\mathcal \{O\}(h+\Delta t)$ in an energy norm. Moreover, the analysis is performed without any assumptions on the speed of propagation of the free boundary and only the realistic regularity $u_t\in \mathcal \{L\}^2(0,T; \mathcal \{L\}^2(\Omega ))$ is assumed. Further, we present some numerical experiments to illustrate the performance of the proposed methods.},
author = {Majumder, Papri},
journal = {Applications of Mathematics},
keywords = {finite element; discontinuous Galerkin method; parabolic obstacle problem},
language = {eng},
number = {5},
pages = {673-699},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem},
url = {http://eudml.org/doc/298218},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Majumder, Papri
TI - Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 5
SP - 673
EP - 699
AB - We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in $\mathbb {R}^d$$(d=2,3)$. For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete schemes, and derive error estimate of optimal order $\mathcal {O}(h+\Delta t)$ in an energy norm. Moreover, the analysis is performed without any assumptions on the speed of propagation of the free boundary and only the realistic regularity $u_t\in \mathcal {L}^2(0,T; \mathcal {L}^2(\Omega ))$ is assumed. Further, we present some numerical experiments to illustrate the performance of the proposed methods.
LA - eng
KW - finite element; discontinuous Galerkin method; parabolic obstacle problem
UR - http://eudml.org/doc/298218
ER -

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