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, $-\sum_{\alpha=1}^d \partial_{x_\alpha} S_{i\alpha}(\nabla u(x)) = f_i(x)$, $i=1,\dots,...
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| Format: | Report |
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2006
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| author | Ortner, C Suli, E |
| author_facet | Ortner, C Suli, E |
| author_sort | Ortner, C |
| collection | OXFORD |
| description | 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. 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). |
| first_indexed | 2024-03-07T00:18:11Z |
| format | Report |
| id | oxford-uuid:7b96ae31-5d6c-4cc7-90e2-2e3a07e63822 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:18:11Z |
| publishDate | 2006 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:7b96ae31-5d6c-4cc7-90e2-2e3a07e638222022-03-26T20:51:32ZDiscontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic SystemsReporthttp://purl.org/coar/resource_type/c_93fcuuid:7b96ae31-5d6c-4cc7-90e2-2e3a07e63822Mathematical Institute - ePrintsUnspecified2006Ortner, CSuli, EWe 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. 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). |
| spellingShingle | Ortner, C Suli, E Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems |
| title | Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems |
| title_full | Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems |
| title_fullStr | Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems |
| title_full_unstemmed | Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems |
| title_short | Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems |
| title_sort | discontinuous galerkin finite element approximation of nonlinear second order elliptic and hyperbolic systems |
| work_keys_str_mv | AT ortnerc discontinuousgalerkinfiniteelementapproximationofnonlinearsecondorderellipticandhyperbolicsystems AT sulie discontinuousgalerkinfiniteelementapproximationofnonlinearsecondorderellipticandhyperbolicsystems |