Ortner, Christoph and Suli, Endre (2006) Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems. Technical Report. Unspecified. (Submitted)
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, , , and , , with , in a bounded spatial domain in , subject to mixed Dirichlet--Neumann boundary conditions, and assuming that is uniformly monotone on . 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 . We then further relax our hypotheses: using a broken Grding inequality we extend our optimal error bounds to the case of quasilinear hyperbolic systems where, instead of assuming that is uniformly monotone, we only require that the fourth-order tensor 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.
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).
|Item Type:||Technical Report (Technical Report)|
|Subjects:||H - N > Numerical analysis|
|Research Groups:||Numerical Analysis Group|
|Deposited By:||Lotti Ekert|
|Deposited On:||11 May 2011 10:56|
|Last Modified:||11 May 2011 10:56|
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