A discontinuous Galerkin recovery scheme with stabilization for diffusion problems
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A discontinuous Galerkin recovery scheme with stabilization for diffusion problems Mauricio Osorio1 · Wilmar Imbachí1 Received: 7 February 2020 / Revised: 11 July 2020 / Accepted: 22 September 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract In this work, ideas previously introduced for a discontinuous Galerkin recovery method in one dimension, that involves a penalty stabilization term, are extended to an elliptic differential equation in several dimensions and different types of boundary conditions and meshes. Using standard arguments for other existing discontinuous Galerkin methods, we show results of existence and uniqueness of the solution. Also, optimal convergence rates are proved theoretically and confirmed numerically. Likewise, the numerical experiments allow us to analyze of the effect of the stabilization parameter. Keywords Discontinuous Galerkin · Polynomial recovery · Error analysis · Diffusion equation Mathematics Subject Classification 65N15 · 65N30 · 65M60
1 Introduction It is well known that there is an important number of numerical methods to approximate the solution of partial differential equations (PDE) of different types. Each of them have its advantages and disadvantages. Finite differences, finite elements and finite volumes are among the more populars and have laid the groundwork for the emergence of new and innovative methods. Discontinuous Galerkin (DG) methods, for instance, have been seen for the computational fluid dynamics (CFD) community as a promising alternative that combines the advantages of finite elements and finite volumes methods: high order accuracy and the possibility of representing discontinuous solutions or discontinuous coefficients. It is, precisely, the lack of continuity that has * Mauricio Osorio [email protected] Wilmar Imbachí [email protected] 1
Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellín, Colombia
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become the most important challenge when modeling diffusion problems and has given rise to the appearance of different DG methods. In [2] Cockburn et al. introduce an unified analysis for most of the existent DG methods until that moment, all of them based on mixed formulations. Polynomial recovery techniques have a long history in finite element methods for the study of superconvergence after a postprocessing step (see [1, 4, 5] for instance). In 2005 (see [18]), Van Leer and collaborators introduced a new DG method, called the Discontinuous Galerkin Recovery method (RDG), that does not fit in the unified analysis given by Cockburn (according to Van Leer) and works over the primal formulation of the differential equation instead of the mixed formulation as most DG methods (LDG, HDG, etc) and thus it reduces the degrees of freedom and computational effort . In this method, the diffuse fluxes are computed using a polynomial recovery function in neighboring cells (similar to the known MUSCL scheme [16]), that in the weak sense is indistinguis
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