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Article

Majumder, Papri
Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem. (English). Applications of Mathematics, vol. 66 (2021), issue 5, pp. 673-699
MSC: 65N15, 65N30 | MR 4299880 | Zbl 07396173 | DOI: 10.21136/AM.2021.0030-20
Keywords:
finite element; discontinuous Galerkin method; parabolic obstacle problem
Summary:
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.
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