Mesh Grading in Isogeometric Analysis

TL;DR

This paper develops graded mesh techniques for isogeometric analysis to effectively approximate solutions with singularities, achieving optimal convergence rates through a multipatch approach and validated by numerical experiments.

Contribution

It introduces a novel multipatch graded mesh construction for singular solutions in isogeometric analysis, ensuring optimal convergence rates.

Findings

Graded meshes improve convergence for singular solutions.

Numerical results confirm theoretical convergence rates.

Mesh grading enhances efficiency in isogeometric analysis.

Abstract

This paper is concerned with the construction of graded meshes for approximating so-called singular solutions of elliptic boundary value problems by means of multipatch discontinuous Galerkin Isogeometric Analysis schemes. Such solutions appear, for instance, in domains with re-entrant corners on the boundary of the computational domain, in problems with changing boundary conditions, in interface problems, or in problems with singular source terms. Making use of the analytic behavior of the solution, we construct the graded meshes in the neighborhoods of such singular points following a multipatch approach. We prove that appropriately graded meshes lead to the same convergence rates as in the case of smooth solutions with approximately the same number of degrees of freedom. Representative numerical examples are studied in order to confirm the theoretical convergence rates and to demonstrate the efficiency of the mesh grading technology in Isogeometric Analysis.