Optimal Order Convergence Implies Numerical Smoothness

zbMATH Open1328.65236arXiv1312.6714MaRDI QIDQ3453553

So-Hsiang Chou

Publication date: 27 November 2015

Abstract: It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential non-convergence and instability.

Full work available at URL: https://arxiv.org/abs/1312.6714

zbMATH Keywords

second-order partial differential equationoptimal-order convergencepiecewise polynomialsnumerical smoothnessacross-interface numerical smoothnessapproximation by piecewise polynomialsdiscontinuous Galerkin solutioninterior numerical smoothnessquasi-uniform family of meshes

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs (65N50)

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