Adaptive least-squares space-time finite element methods

TL;DR

This paper introduces an adaptive least-squares space-time finite element method for solving second order linear PDEs, providing a framework for error estimation and adaptive refinement in time-dependent problems.

Contribution

It develops a novel least-squares approach with a saddle point formulation and adaptive error indicators for efficient space-time finite element solutions.

Findings

Derives a priori error estimates based on discrete inf-sup stability.

Uses the adjoint variable as an a posteriori error indicator.

Demonstrates applicability to Poisson, heat, and wave equations.

Abstract

We consider the numerical solution of an abstract operator equation $Bu=f$ by using a least-squares approach. We assume that $B: X \to Y^*$ is an isomorphism, and that $A : Y \to Y^*$ implies a norm in $Y$, where $X$ and $Y$ are Hilbert spaces. The minimizer of the least-squares functional $\frac{1}{2} \, \| Bu-f \|_{A^{-1}}^2$, i.e., the solution of the operator equation, is then characterized by the gradient equation $Su=B^* A^{-1}f$ with an elliptic and self-adjoint operator $S:=B^* A^{-1} B : X \to X^*$. When introducing the adjoint $p = A^{-1}(f-Bu)$ we end up with a saddle point formulation to be solved numerically by using a mixed finite element method. Based on a discrete inf-sup stability condition we derive related a priori error estimates. While the adjoint $p$ is zero by construction, its approximation $p_h$ serves as a posteriori error indicator to drive an adaptive scheme when discretized appropriately. While this approach can be applied to rather general equations, here we consider second order linear partial differential equations, including the Poisson equation, the heat equation, and the wave equation, in order to demonstrate its potential, which allows to use almost arbitrary space-time finite element methods for the adaptive solution of time-dependent partial differential equations.