Solver algorithm for stabilized space-time formulation of advection-dominated diffusion problem

TL;DR

This paper develops an efficient, stabilized solver for a space-time formulation of advection-dominated diffusion problems, using high-order B-spline basis functions and a least-squares approach to avoid oscillations.

Contribution

It introduces a novel solver for a stabilized, symmetric space-time formulation employing a least-squares approach with high-order B-spline discretizations.

Findings

The solver effectively avoids artificial oscillations in advection-dominated problems.

It demonstrates the use of adaptive space-time finite element methods.

Numerical examples validate the efficiency and stability of the proposed approach.

Abstract

This article shows how to develop an efficient solver for a stabilized numerical space-time formulation of the advection-dominated diffusion transient equation. At the discrete space-time level, we approximate the solution by using higher-order continuous B-spline basis functions in its spatial and temporal dimensions. This problem is very difficult to solve numerically using the standard Galerkin finite element method due to artificial oscillations present when the advection term dominates the diffusion term. However, a first-order constraint least-square formulation allows us to obtain numerical solutions avoiding oscillations. The advantages of space-time formulations are the use of high-order methods and the feasibility of developing space-time mesh adaptive techniques on well-defined discrete problems. We develop a solver for a least-square formulation to obtain a stabilized and symmetric problem on finite element meshes. The computational cost of our solver is bounded by the cost of the inversion of the space-time mass and stiffness (with one value fixed at a point) matrices and the cost of the GMRES solver applied for the symmetric and positive definite problem. We illustrate our findings on an advection-dominated diffusion space-time model problem and present two numerical examples: one with isogeometric analysis discretizations and the second one with an adaptive space-time finite element method.