[Paper Review] Finite Element Exterior Calculus for Evolution Problems

Andrew Gillette, Michael Holst+1 more|arXiv (Cornell University)|Feb 8, 2012

Advanced Numerical Methods in Computational Mathematics17 references3 citations

TL;DR

This paper extends the Finite Element Exterior Calculus (FEEC) framework to parabolic and hyperbolic evolution problems by combining Hilbert complex theory with semi-discrete Galerkin finite element methods in time-parameterized Bochner spaces. It establishes optimal a priori error estimates in natural parametrized Hilbert norms, generalizing classical results by Thomée and Geveci to arbitrary spatial dimensions and mixed finite element families, while extending to semi-linear evolution systems via the Holst-Stern framework.

ABSTRACT

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281--354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12:3 (2012), 263--293 and 363--387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomee for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomee and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

Motivation & Objective

• To extend the Finite Element Exterior Calculus (FEEC) framework from elliptic to parabolic and hyperbolic evolution problems.
• To generalize classical a priori error estimates by Thomée (parabolic) and Geveci (hyperbolic) to arbitrary spatial dimensions and mixed finite element methods.
• To incorporate variational crimes and semi-linear terms into the FEEC framework for evolution systems, building on Holst and Stern’s prior work.
• To establish error estimates in time-parameterized Bochner spaces using the structure of Hilbert complexes and de Rham complexes.
• To demonstrate that the FEEC framework ensures stability and convergence for geometric and nonlinear evolution problems on Riemannian manifolds.

Proposed method

• Formulates evolution problems as time-parameterized Hilbert spaces (Bochner spaces) to apply FEEC tools to time-dependent PDEs.
• Applies the FEEC framework via Hilbert complexes and de Rham complexes to mixed variational formulations of evolution systems.
• Uses semi-discrete Galerkin finite element methods, discretizing only in space while preserving time-continuity.
• Employs abstract error estimates from Holst and Stern, adapted to evolution problems through parametrized norms and time-dependent projections.
• Borrows and extends the abstract error estimate (A.1) from Holst and Stern [21], simplifying it for the case of k=n and eliminating harmonic forms.
• Applies optimal approximation properties of finite element projections π_h^k to bound error terms in terms of Sobolev norms of the exact solution.

Experimental results

Research questions

• RQ1 Can the FEEC framework be extended to handle parabolic and hyperbolic evolution problems using Hilbert complex theory?
• RQ2 Do the classical a priori error estimates of Thomée and Geveci generalize to arbitrary spatial dimensions and higher-order mixed finite element methods?
• RQ3 Can the FEEC framework accommodate variational crimes and semi-linear terms in evolution problems?
• RQ4 What is the role of the de Rham complex and Bochner space structure in deriving error estimates for time-dependent mixed finite element methods?
• RQ5 How do the coefficients η, δ, μ in the abstract error estimate influence convergence rates in evolution problems?

Key findings

• The paper establishes optimal a priori error estimates for Galerkin FEM approximations of parabolic and hyperbolic evolution problems in Bochner space norms.
• The error estimate for the solution u in L² norm is bounded by c(E(u) + ηE(σ) + (δ+μ)E(dσ) + μE(P_B u)), with E(w) denoting best approximation error in W-norm.
• For the case k=n in the de Rham complex, the estimate simplifies to ||u - ũ_h||_L² ≤ c(E(u) + ηE(σ) + (δ+μ)E(dσ) + μE(P_B u)) by eliminating harmonic and exact forms.
• The convergence rate is shown to be O(h^{s+2}) for s ≤ r+1, where s is the Sobolev regularity and r is the polynomial degree of the finite element space.
• The coefficients η=O(h), δ=O(h^{min(2,r+1)}), and μ=O(h^{r+1}) are derived from abstract operator norms in the de Rham context.
• The results recover and generalize Thomée and Geveci’s estimates for 2D domains and lowest-order mixed methods as special cases.