Adaptive discontinuous Galerkin approximations to fourth order parabolic problems

TL;DR

This paper develops an adaptive numerical scheme combining implicit Euler and discontinuous Galerkin methods for fourth order parabolic problems, with error estimates guiding efficient computations in convex domains.

Contribution

It introduces an a posteriori error analysis for adaptive algorithms applied to high-order parabolic problems, improving computational efficiency.

Findings

Significant reduction in computational effort through adaptivity.

Effective a posteriori error estimates in $L^{ abla}(L^2)$ and $L^2(L^2)$ norms.

Validated the approach for convex domains in 2D and 3D.

Abstract

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r\ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort.