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Discontinuous Galerkin Methods for a Class of Nonvariational Problems

Communications on Applied Mathematics and Computation 2022年第2期4卷 634-656页

作者： andreas dedner Tristan Pryer Mathematics Institute University of WarwickCoventry CV47ALUK Department of Mathematical Sciences University of BathBath BA27AYUK

We extend the fnite element method introduced by Lakkis and Pryer(SIAM ***.33(2):786–801,2011)to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin(DG)*** is done by viewing the“fnite element Hessian”as an auxiliary variable in the *** the fnite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic ***,the system matrix is very easy to assemble;thus,this approach greatly reduces the computational complexity of the discretisation compared to the continuous *** conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al.(SIAM ***.39(5):1749–1779,2001/2002).We also give an a posteriori analysis of the method in the case where the problem has a strong *** analysis applies to any consistent representation of the fnite element Hessian,and thus is applicable to the previous works making use of continuous Galerkin *** evidence is presented showing that the method works well also in a more general setting.

关键词： Nonvariational problems Discontinuous Galerkin Error estimates Adaptivity

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Extendible and Efcient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods

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Communications on Applied Mathematics and Computation 2022年第2期4卷 657-696页

作者： andreas dedner Robert Klöfkorn University of Warwick Coventry CV47ALUK Lund University Box 11822100 LundSweden

This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efcient implementations of the discontinuous Galerkin(DG)method for solving a wide range of nonlinear partial diferential equations(PDEs).Although the C++interfaces of Dune-Fem-DG are highly fexible and customizable,a solid knowledge of C++is necessary to make use of this powerful *** this work,easier user interfaces based on Python and the unifed form language are provided to open Dune-Fem-DG for a broader *** Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs.

关键词： Dune Dune-Fem Discontinuous Galerkin Finite volume Python Advection-difusion Euler Navier-Stokes

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