Volume 65, pp. 387-413, 2026.

Adaptive least-squares space-time finite element methods

Christian Köthe, Richard Löscher, and Olaf Steinbach

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 $S\coloneqq B^* A^{-1} B : X \to X^*$. 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 an 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 one to use almost arbitrary space-time finite element meshes for the adaptive solution of time-dependent partial differential equations.

Full Text (PDF) [1.6 MB], BibTeX , DOI: 10.1553/etna_vol65s387

Key words

least-squares methods, space-time finite element methods, a posteriori error indicator, adaptivity, Poisson equation, heat equation, wave equation

AMS subject classifications

65M60, 65M12, 65M50, 65N30, 65N12, 65N50

Links to the cited ETNA articles

[51]	Vol. 52 (2020), pp. 154-194 Olaf Steinbach and Marco Zank: Coercive space-time finite element methods for initial boundary value problems