Discontinuous Galerkin Isogeometric Analysis of Elliptic Diffusion Problems on Segmentations with Gaps

TL;DR

This paper introduces a novel discontinuous Galerkin Isogeometric Analysis method for elliptic diffusion problems on complex domains with gaps, providing a rigorous framework and demonstrating effectiveness through numerical examples.

Contribution

It develops a new IgA technique that handles non-matching interfaces and gap regions, extending the applicability of IgA to more complex domain decompositions.

Findings

Method achieves optimal convergence rates in numerical tests.

Theoretical analysis shows gap size affects convergence.

Numerical examples confirm the robustness of the approach.

Abstract

We propose a new discontinuous Galerkin Isogeometric Analysis (IgA) technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with non-matching interfaces. Due to an incorrect segmentation procedure, it may happen that the interfaces of adjacent subdomains don't coincide. In this way, gap regions, which are not present in the original physical domain, are created. In this paper, the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains. We apply a multi-patch approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem. The last formulation is further modified by deriving interface conditions without the presence of the solution in the gap. Finally, the solution of this modified problem is approximated by a special discontinuous Galerkin IgA technique. The ideas are illustrated on a model diffusion problem with discontinuous diffusion coefficients. We develop a rigorous theoretical framework for the proposed method clarifying the influence of the gap size onto the convergence rate of the method. The theoretical estimates are supported by numerical examples in two- and three-dimensional computational domains.