A class of Discontinuous Galerkin methods for nonlinear variational problems

TL;DR

This work develops a class of elementwise nonconforming DG methods for nonlinear variational problems with convex energies, formulating an energy-based discretisation and proving convergence of discrete minimisers to the unique continuous minimiser as the mesh is refined. The authors introduce discrete energies $\mathcal{E}_{dg}$ that couple elementwise gradients with interface terms via a lifting-based discrete gradient, and they add penalty terms to guarantee coercivity and stability. They establish Γ-convergence of the discrete energies to the continuous energy $\mathcal{E}$ under convexity of $W$ and growth assumptions, and they derive a priori error estimates in the smooth regime using fixed-point arguments and generalized Gårding inequalities. Numerical experiments corroborate the theory and illustrate the robustness of the penalty formulations, extending DG techniques to nonlinear elasticity-type variational problems with convex energies. The framework provides a rigorous, energy-consistent DG approach for nonlinear variational problems, with potential impact on reliable simulations in nonlinear elasticity and related applications.

Abstract

In the context of Discontinuous Galerkin methods, we study approximations of nonlinear variational problems associated with convex energies. We propose element-wise nonconforming finite element methods to discretize the continuous minimisation problem. Using $Γ$-convergence arguments we show that the discrete minimisers converge to the unique minimiser of the continuous problem as the mesh parameter tends to zero, under the additional contribution of appropriately defined penalty terms at the level of the discrete energies. We finally substantiate the feasibility of our methods by numerical examples.

Figures (4)

• Figure 1: Comparing the accuracy of the two proposed stabilisation penalty terms \ref{['eq:penalty']} (circles) and \ref{['eq:penalty2']} (squares) for the convex strain energies $W(F) = |F|^4$ and $W(F) = |F|^6$. For the former energy the material is subjected to 10% uniaxial tension, while for the latter to 10% uniaxial compression. Horizontal axis: values of the penalty parameter $\alpha$. Vertical axes: $|u_h - u_e|_{L^1(\Omega)}$ and $|u_h - u_e|_{W^{1,1}(\Omega)}$ errors for various mesh resolutions, specifically for 1024 (blue), 2034 (green) and 4096 (orange) triangles, where $u_h$ denotes the numerical solutions and $u_e$ the exact homogeneous minimizers. In \ref{['fig:fn1a']} and \ref{['fig:fn1c']} circular error belong in the range of $10^{-8}$ and $10^{-9}$, while in \ref{['fig:fn1b']} and \ref{['fig:fn1d']} circular errors are of the order $10^{-6}$, which is determined by the numerical minimization criterion, i.e. stop iterations when $\sup_{K \in T_h}\lVert\nabla u_h\rVert_{L^{\infty}(K)} < 10^{-5}.$ In \ref{['fig:fn1c']} and \ref{['fig:fn1d']} the scheme employing the penalty \ref{['eq:penalty2']} did not converge for the values $\alpha =20$ and $40$.
• Figure 2: Reference to deformed triangle area ($\det \nabla{y_h}$) in the reference configuration under uniaxial tension (10% strain). Employing the penalty of eq. \ref{['eq:penalty2']}, $\det \nabla{y_h}$ for the computed minimizers is illustrated: \ref{['fig:convex_a1_td']} when $\alpha=20$ and \ref{['fig:convex_a4_td']} when $\alpha =160$ (see blue squares of Figs. \ref{['fig:fn1a']} and \ref{['fig:fn1b']} at $\alpha =20, 160$). \ref{['fig:quasiconvex_a1_td']}: Area ratio for the penalty of eq.\ref{['eq:penalty']} with $\alpha=20$. \ref{['fig:exact_td']}$\det \nabla{I_h y_0}$, $I_h:W^{1,p}(\Omega) \rightarrow V_h$, the standard interpolation operator.
• Figure 3: Convergence behaviour in $W^{1,2}(\Omega,T_h)$ of the error over discretization parameter $h.$ Table 1 and plot (a) correspond to $p=2$ and Table 2 and plot (b) correspond to $p=4$ in (\ref{['eq:error_estimates']}) for various values of the penalty parameter $a.$
• Figure :

Theorems & Definitions (19)

• Definition 2.1
• Lemma 3.1: Bound on global lifting operator
• Lemma 3.2
• Lemma 3.3: Poincaré inequality for DG spaces.
• proof
• Theorem 3.1
• Lemma 3.4
• proof
• Remark 3.1
• Lemma 3.5