A sparse-grid isogeometric solver

TL;DR

This paper explores the application of sparse-grid techniques to Isogeometric Analysis (IGA), demonstrating potential benefits in efficiency and parallelization, especially for smooth solutions or when singularities are known.

Contribution

It introduces a sparse-grid IGA solver, analyzing its effectiveness for smooth and non-smooth PDE solutions and highlighting its advantages in parallel computing.

Findings

Sparse-grid IGA is effective for smooth PDE solutions.

Knowledge of solution singularities improves sparse-grid performance.

Sparse grids facilitate straightforward parallelization of IGA solvers.

Abstract

Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.