[Paper Review] Space-time least-squares finite elements for parabolic equations

Thomas Führer, Michael Karkulik|arXiv (Cornell University)|Nov 5, 2019

Advanced Numerical Methods in Computational Mathematics2 citations

TL;DR

This paper introduces a space-time least-squares finite element method for the heat equation based on minimizing an L2 residual functional over a first-order system formulation. The approach ensures uniform stability, symmetric positive definite linear systems, and enables full space-time adaptivity with built-in a-posteriori error estimators, achieving optimal convergence rates in numerical experiments on simplicial meshes.

ABSTRACT

We present a space-time least squares finite element method for the heat equation. It is based on residual minimization in L2 norms in space-time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have a local a-posteriori error estimator for free. In particular, our approach features full space-time adaptivity. We also present a-priori error analysis on simplicial space-time meshes which are highly structured. Numerical results conclude this work.

Motivation & Objective

• To develop a stable, conforming space-time finite element method for parabolic PDEs that avoids the limitations of time-stepping schemes.
• To ensure uniform stability for any conforming discrete space by leveraging least-squares variational formulation.
• To enable full space-time adaptivity through local error indicators derived from the least-squares functional.
• To provide a priori error estimates on structured simplicial space-time meshes.
• To demonstrate optimal convergence rates and robust performance in numerical experiments across 1D and 2D problems.

Proposed method

• Formulate the heat equation as a first-order system: ∂tu − div σ = f, σ − ∇u = 0, with initial condition u(0) = u₀.
• Define the least-squares functional j(u, σ) = ∫₀ᵀ ‖∂tu − div σ − f‖²_{L²(Ω)} dt + ∫₀ᵀ ‖σ − ∇u‖²_{L²(Ω)} dt + ‖u(0) − u₀‖²_{L²(Ω)}.
• Minimize j(u, σ) over a product space of H¹(0,T;L²(Ω)) × L²(0,T;H¹(Ω)) with appropriate trace and regularity constraints.
• Derive a symmetric, coercive bilinear form from the functional, ensuring stable and sparse stiffness matrices in the discrete setting.
• Construct local a-posteriori error estimators by decomposing the functional into element-wise contributions.
• Implement adaptive refinement driven by the error estimator, allowing local space-time mesh enrichment.

Experimental results

Research questions

• RQ1 Can a space-time least-squares finite element method achieve uniform stability for arbitrary conforming discrete spaces in parabolic problems?
• RQ2 Does residual minimization in L² norms over space-time yield symmetric, positive definite, and sparse algebraic systems?
• RQ3 Can the least-squares functional be used to derive reliable and efficient local error estimators for adaptive refinement?
• RQ4 What convergence rates are achieved by the method on simplicial space-time meshes for problems with varying regularity?
• RQ5 How does adaptive refinement compare to uniform refinement in terms of convergence rates and computational efficiency?

Key findings

• The method achieves uniform stability independent of the discrete space, ensuring robustness for arbitrary conforming discretizations.
• The resulting stiffness matrix is symmetric, positive definite, and sparse, enabling efficient solution with iterative solvers like preconditioned CG.
• A-posteriori error estimators derived from the least-squares functional are naturally local and enable effective space-time adaptivity.
• Numerical experiments show optimal convergence rates of approximately 0.45 for adaptive refinement in 1D, outperforming uniform refinement with rate 0.25.
• In 2D problems with singularities (e.g., reentrant corners), adaptive refinement improves convergence rates from ~0.2 (uniform) to ~0.24, demonstrating effectiveness in handling reduced regularity.
• For smooth solutions, the method achieves convergence rates of order N⁻¹/² in the L² norm, consistent with optimal h-refinement behavior.