Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems

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, − ∑d α=1 ∂xαSiα(∇u(x)) = fi(x), i = 1, . . . , d, and ∂2 t ui− ∑d α=1 ∂xαSiα(∇u(t, x)) = fi(t, x), i = 1, . . . , d, with ∂xα = ∂/∂xα, in a bounded spatial domain in R d, subject to mixed Dirichlet–Neumann boundary conditions, and assuming that S = (Siα) is uniformly monotone on R d×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 G̊arding 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 = ∇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.