Space-time least-squares isogeometric method and efficient solver for parabolic problems

TL;DR

This paper introduces a space-time least-squares isogeometric method for parabolic problems that leverages high-degree splines and an efficient Sylvester-equation-based preconditioner, achieving high computational efficiency and parallelization potential.

Contribution

The paper presents a novel space-time least-squares isogeometric approach with a robust, efficient solver using tensor-product splines and Sylvester-like equations, improving computational performance for parabolic problems.

Findings

Preconditioner performance is robust to spline degree and mesh size.

Application time is nearly proportional to degrees-of-freedom, independent of polynomial degree.

Method is suitable for parallel computing environments.

Abstract

In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degrees-of-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization.