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.
Highlights
Fourth order parabolic equations and corresponding initial-boundary value problems appear in the modelling in areas as diverse as biology, phase-field modelling and image processing to name a few
An adaptive algorithm, based on residual type a posteriori indicators of errors measured in L∞(L2) and L2(L2) 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 simple model problem (1)–(3) appears to be sufficient in highlighting some of the challenges in the error estimation and adaptivity of finite element methods for more complex fourth order parabolic problems. It appears that the derived a posteriori bounds and the respective adaptive algorithms can be modified in a straightforward fashion to include the original discontinuous Galerkin (dG) method of Baker [3] and C0-interior penalty methods [9,17]
Summary
Fourth order parabolic equations and corresponding initial-boundary value problems appear in the modelling in areas as diverse as biology, phase-field modelling and image processing to name a few. A posteriori error estimates in an L2(H2)-type norm and adaptive algorithms for fully discrete schemes with discontinuous Galerkin methods for fourth order problems are proposed in [43]. This work is concerned with the derivation of a posteriori error estimates in weaker than L2(H2)-norms and their use within an adaptive algorithm for a class of discontinuous Galerkin interior penalty methods for a fully discrete approximation of the problem:. The simple model problem (1)–(3) appears to be sufficient in highlighting some of the challenges in the error estimation and adaptivity of finite element methods for more complex fourth order parabolic problems It appears that the derived a posteriori bounds and the respective adaptive algorithms can be modified in a straightforward fashion to include the original dG method of Baker [3] and C0-interior penalty methods [9,17].
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