Analysis of Multipatch Discontinuous Galerkin IgA Approximations to Elliptic Boundary Value Problems

TL;DR

This paper analyzes the approximation properties of multi-patch dG-IgA methods for elliptic boundary value problems, demonstrating optimal convergence rates despite discontinuities across patch interfaces.

Contribution

It provides a rigorous a priori error analysis for multi-patch dG-IgA methods applied to elliptic problems with discontinuous coefficients in 2D and 3D.

Findings

Optimal convergence rates achieved in dG-norm

Discontinuous solutions across patch interfaces handled effectively

Error estimates valid for solutions in certain Sobolev spaces

Abstract

In this work, we study the approximation properties of multi-patch dG-IgA methods, that apply the multipatch Isogeometric Analysis (IgA) discretization concept and the discontinuous Galerkin (dG) technique on the interfaces between the patches, for solving linear diffusion problems with diffusion coefficients that may be discontinuous across the patch interfaces. The computational domain is divided into non-overlapping sub-domains, called patches in IgA, where $B$-splines, or NURBS finite dimensional approximations spaces are constructed. The solution of the problem is approximated in every sub-domain without imposing any matching grid conditions and without any continuity requirements for the discrete solution across the interfaces. Numerical fluxes with interior penalty jump terms are applied in order to treat the discontinuities of the discrete solution on the interfaces. We provide a rigorous a priori discretization error analysis for problems set in 2d- and 3d- dimensional domains, with solutions belonging to $W^{l,p}, l\geq 2,{\ } p\in ({2d}/{(d+2(l-1))},2]$. In any case, we show optimal convergence rates of the discretization with respect to the dG - norm.