A quasi-robust discretization error estimate for discontinuous Galerkin Isogeometric Analysis

TL;DR

This paper provides a quasi-robust error estimate for discontinuous Galerkin Isogeometric Analysis, enabling more efficient multigrid solvers for complex, non-conforming geometries with high-order spline discretizations.

Contribution

It introduces error estimates that depend only poly-logarithmically on the spline degree, facilitating the development of fast solvers for non-conforming multipatch IGA.

Findings

Error estimates with poly-logarithmic dependence on spline degree.

Enhanced potential for multigrid solver development.

Applicability to non-conforming multipatch geometries.

Abstract

Isogeometric Analysis is a spline-based discretization method to partial differential equations which shows the approximation power of a high-order method. The number of degrees of freedom, however, is as small as the number of degrees of freedom of a low-order method. This does not come for free as the original formulation of Isogeometric Analysis requires a global geometry function. Since this is too restrictive for many kinds of applications, the domain is usually decomposed into patches, where each patch is parameterized with its own geometry function. In simpler cases, the patches can be combined in a conforming way. However, for non-matching discretizations or for varying coefficients, a non-conforming discretization is desired. An symmetric interior penalty discontinuous Galerkin (SIPG) method for Isogeometric Analysis has been previously introduced. In the present paper, we give error estimates that only depend poly-logarithmically on the spline degree. This opens the door towards the construction and the analysis of fast linear solvers, particularly multigrid solvers for non-conforming multipatch Isogeometric Analysis.