Numerical Analysis Group Research Report NA-06/05

May 2006, 49 pages

We develop the convergence analysis of discontinuous Galerkin finite element approximations to second-order quasilinear elliptic and hyperbolic systems of partial differential equations of the form, respectively, $-\sum_{\alpha=1}^d \partial_{x_\alpha} S_{i\alpha}(\nabla u(x)) = f_i(x)$, $i=1,\dots, d$, and $\partial^2_t{u}_i-\sum_{\alpha=1}^d \partial_{x_\alpha} S_{i\alpha}(\nabla u(t,x)) = f_i(t,x)$, $i=1,\dots, d$, with $\partial_{x_\alpha} = \partial/\partial x_\alpha$, in a bounded spatial domain in $\mathbb{R}^d$, subject to mixed Dirichlet--Neumann boundary conditions, and assuming that $S=(S_{i\alpha})$ is uniformly monotone on $\mathbb{R}^{d\times d}$. The associated energy functional is then uniformly convex. An optimal order bound is derived on the discretization error in each case without requiring the global Lipschitz continuity of the tensor $S$. We then further relax our hypotheses: using a broken G{\aa}rding inequality we extend our optimal error bounds to the case of quasilinear hyperbolic systems where, instead of assuming that $S$ is uniformly monotone, we only require that the fourth-order tensor $A=\nabla S$ is satisfies a Legendre--Hadamard condition. The associated energy functional is then only rank-1 convex. Evolution problems of this kind arise as mathematical models in nonlinear elastic wave propagation.

Key words and phrases: Nonlinear elliptic and hyperbolic systems of partial differential equations, discontinuous Galerkin methods, Legendre-Hadamard condition, broken Garding inequality.

The authors acknowledge the financial support received from the European research project HPRN-CT-2002-00284: New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation, and the kind hospitality of Carlo Lovadina and Matteo Negri (University of Pavia).

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