Abstract
We address the question of the rates of convergence of the p-version interior penalty discontinuous Galerkin method (p-IPDG) for second order elliptic problems with non-homogeneous Dirichlet boundary conditions. It is known that the p-IPDG method admits slightly suboptimal a-priori bounds with respect to the polynomial degree (in the Hilbertian Sobolev space setting). An example for which the suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and validated in practice through numerical experiments is presented. Moreover, the performance of p-IPDG on the related problem of p-approximation of corner singularities is assessed both theoretically and numerically, witnessing an almost doubling of the convergence rate of the p-IPDG method.
Highlights
Discontinuous Galerkin (DG) methods for elliptic problems have gained popularity in recent years
To the best of our knowledge, the sharpest known general error bounds for the hp-version interior penalty DG method for second-order elliptic PDEs are due to Riviere, Wheeler and Girault [11] and Houston, Schwab and Suli [9]; when the error is measured in the energy norm, the a-priori bounds are optimal with respect to the meshsize h but are suboptimal with respect to the polynomial degree p by half an order of p
We focus on the p-version interior penalty discontinuous Galerkin finite element method (p-IPDG), addressing the question of the rates of convergence for second order elliptic problems with non-homogeneous Dirichlet boundary conditions, when the underlying analytical solution belongs to a Hilbertian Sobolev space Hk
Summary
Discontinuous Galerkin (DG) methods for elliptic problems have gained popularity in recent years. Optimal error bounds for the hp-version interior penalty DG method are known in the case where the underlying discontinuous Galerkin finite element space admits an H1-conforming subspace of the same polynomial order up to the boundary; see, e.g., [6, Theorem 8.2] and the subsequent discussion therein. We focus on the p-version interior penalty discontinuous Galerkin finite element method (p-IPDG), addressing the question of the rates of convergence for second order elliptic problems with non-homogeneous Dirichlet boundary conditions, when the underlying analytical solution belongs to a (standard) Hilbertian Sobolev space Hk. we present an example for which the suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and validated through numerical experiments; the known a-priori bounds from the literature [11, 9] are sharp, i.e., the p-IPDG method is suboptimal by half an order of p.
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