Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces

TL;DR

This paper introduces a novel Discontinuous Galerkin isogeometric analysis method for solving elliptic PDEs on surfaces, allowing for domain decomposition with discontinuities only at patch boundaries, supported by theoretical analysis and numerical validation.

Contribution

It presents the first DG method tailored for isogeometric analysis on surfaces, enabling discontinuities at patch boundaries with rigorous error analysis.

Findings

The DG scheme achieves optimal discretization error rates.

Numerical results confirm the theoretical error estimates.

Method effectively handles complex surface geometries.

Abstract

Isogeometric analysis uses the same class of basis functions for both, representing the geometry of the computational domain and approximating the solution. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This patch representation corresponds to a domain decomposition. In this paper, we present for the first time a Discontinuous Galerkin (DG) Method that allows for discontinuities only along the sub-domain (patch) boundaries. The required smoothness is obtained by the DG terms associated with the boundary of the sub-domains. The construction and corresponding discretization error analysis of such DG scheme is presented for Elliptic PDEs in a 2D as well as on open and closed surfaces. Furthermore, we present numerical results to confirm the theory presented.