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Register a new userSuperconvergence of Discontinuous Galerkin methods for Elliptic Boundary Value Problems
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Added by
Limin Ma
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Limin Ma
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In this paper, we present a unified analysis of the superconvergence property for a large class of mixed discontinuous Galerkin methods. This analysis applies to both the Poisson equation and linear elasticity problems with symmetric stress formulations. Based on this result, some locally postprocess schemes are employed to improve the accuracy of displacement by order min(k+1, 2) if polynomials of degree k are employed for displacement. Some numerical experiments are carried out to validate the theoretical results.
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This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations. By the technique of constructing some special correction functions, we prove the $(2k+1)$th order superconvergence for the cell averages, and the numerical traces in the discrete $L^2$ norm. In addition, superconvergence of order $k+2$ and $k+1$ are obtained for the error and its derivative at generalized Radau points. All theoretical findings are confirmed by numerical experiments.
We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the $h$- and $
In this manuscript we present an approach to analyze the discontinuous Galerkin solution for general quasilinear elliptic problems. This approach is sufficiently general to extend most of the well-known discretization schemes, including BR1, BR2, SIPG and LDG, to nonlinear cases in a canonical way, and to establish the stability of their solution. Furthermore, in case of monotone and globally Lipschitz problems, we prove the existence and uniqueness of the approximated solution and the $h$-optimality of the error estimate in the energy norm as well as in the $L_2$ norm.
We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre-Gauss-Lobatto quadrature. The theoretical discussion serves to re-contextualize stable filtering results for finite difference methods into the DG setting. It is shown that the stability of the filtering is equivalent to a particular contractivity condition borrowed from the analysis of so-called transmission problems. As such, the temporal stability proof relies on the fact that the underlying spatial discretization of the problem possesses a semi-discrete bound on the solution. Numerical tests are provided to verify and validate the underlying theoretical results.
A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different discretization variables, $u_h, bm{p}_h, check u_h$ and $check p_h$, where $u_h$ and $bm{p}_h$ are for approximation of $u$ and $bm{p}=-alpha abla u$ inside each element, and $ check u_h$ and $check p_h$ are for approximation of residual of $u$ and $bm{p} cdot bm{n}$ on the boundary of each element. The resulting 4-field discretization is proved to satisfy inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, most existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.
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