[Paper Review] $C^0$ discontinuous Galerkin finite element methods for second order linear elliptic partial differential equations in non-divergence form

Xiaobing Feng, Lauren Hennings+1 more|arXiv (Cornell University)|May 12, 2015

Advanced Numerical Methods in Computational Mathematics8 references4 citations

TL;DR

This paper introduces a $C^0$ discontinuous Galerkin finite element method for second-order linear elliptic PDEs in non-divergence form with continuous coefficients. By incorporating an interior penalty term into a nonsymmetric, PDE-induced bilinear form, the method achieves optimal convergence in a discrete $W^{2,p}$ norm for polynomial degrees $k \geq 2$, proven via a discrete Calderón–Zygmund estimate and mimicking strong solution techniques at the discrete level.

ABSTRACT

This paper is concerned with finite element approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. A nonstandard (primal) finite element method, which uses finite-dimensional subspaces consisting globally continuous piecewise polynomial functions, is proposed and analyzed. The main novelty of the finite element method is to introduce an interior penalty term, which penalizes the jump of the flux across the interior element edges/faces, to augment a nonsymmetric piecewise defined and PDE-induced bilinear form. Existence, uniqueness and error estimate in a discrete $W^{2,p}$ energy norm are proved for the proposed finite element method. This is achieved by establishing a discrete Calderon-Zygmund-type estimate and mimicking strong solution PDE techniques at the discrete level. Numerical experiments are provided to test the performance of proposed finite element method and to validate the convergence theory.

Motivation & Objective

• To develop a stable, convergent finite element method for second-order elliptic PDEs in non-divergence form, where standard Galerkin methods fail due to lack of divergence structure.
• To address the challenge of approximating $W^{2,p}$ strong solutions when the coefficient matrix $A$ is continuous but not differentiable, precluding weak formulation approaches.
• To design a primal finite element method using globally continuous piecewise polynomials that is computationally simple and implementable in standard FEM software.
• To establish optimal convergence rates in a discrete $W^{2,p}$ energy norm through novel discrete stability estimates resembling PDE-level strong solution techniques.
• To validate the method numerically across problems with smooth, uniformly continuous, and degenerate coefficients, demonstrating convergence even beyond theoretical assumptions.

Proposed method

• The method employs a primal formulation using $C^0$ finite element spaces of globally continuous piecewise polynomials of degree $k \geq 2$.
• A nonsymmetric bilinear form is constructed that incorporates a PDE-induced term and an interior penalty term penalizing jumps in the flux across element interfaces.
• The interior penalty term enforces weak continuity of the gradient, mimicking features of interior penalty DG methods while preserving $C^0$ continuity of the discrete solution.
• Stability and convergence are established via a discrete Calderón–Zygmund-type estimate, derived by treating the local discretization as a perturbation of a constant-coefficient divergence-form operator.
• The analysis follows strong solution techniques from PDE theory, particularly those involving local stability and global Gärding-type inequalities in non-divergence form.
• The method is implemented using standard finite element software, with the bilinear form assembled element-wise and the penalty parameter chosen to ensure stability.

Experimental results

Research questions

• RQ1 Can a stable and convergent $C^0$ finite element method be constructed for second-order elliptic PDEs in non-divergence form with continuous coefficients?
• RQ2 Does introducing an interior penalty term to a nonsymmetric, PDE-induced bilinear form yield optimal convergence in a discrete $W^{2,p}$ norm?
• RQ3 Can discrete Calderón–Zygmund estimates be established for non-divergence form PDEs to mirror the stability theory of strong solutions?
• RQ4 Is the method robust for problems with non-uniformly elliptic or degenerate coefficients, even when theoretical convergence is not guaranteed?
• RQ5 What convergence rates are observed in practice for $H^1$ and piecewise $H^2$ errors across varying polynomial degrees and problem types?

Key findings

• The proposed $C^0$ DG finite element method achieves optimal convergence in the discrete $W^{2,p}$ norm for polynomial degrees $k \geq 2$ on quasi-uniform meshes.
• Numerical experiments confirm that $|u - u_h|_{H^1( Omega)} = \mathcal{O}(h^k)$ and $\|D_h^2(u - u_h)\|_{L^2( Omega)} = \mathcal{O}(h^{k-1})$, matching theoretical predictions for smooth problems.
• For a problem with $W^{2,p}$ solution and $p < 8/5$, the method converges with rate $\mathcal{O}(h^{3/4 - \varepsilon})$ in the piecewise $H^2$ norm when $k \geq 2$, consistent with theory.
• Even for a degenerate coefficient matrix with $\det(A) = 0$, the method exhibits convergence with observed rates $\|u - u_h\|_{L^2} = \mathcal{O}(h^{4/3})$ and $|u - u_h|_{H^1} = \mathcal{O}(h^{5/6})$, suggesting robustness beyond current theory.
• The $H^1$ error converges optimally with order $\mathcal{O}(h^k)$, and the method remains convergent even in the piecewise linear case ($k=1$), though optimal rates are only proven for $k \geq 2$.
• The method is structurally simple, computationally efficient, and compatible with standard finite element software, enabling straightforward implementation.