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...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 tool.With this work,easier user interfaces based on Python and the unifed form language are provided to open Dune-Fem-DG for a broader audience.The Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs.展开更多
We extend the fnite element method introduced by Lakkis and Pryer(SIAM J.Sci.Comput.33(2):786–801,2011)to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontin...We extend the fnite element method introduced by Lakkis and Pryer(SIAM J.Sci.Comput.33(2):786–801,2011)to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin(DG)framework.This is done by viewing the“fnite element Hessian”as an auxiliary variable in the formulation.Representing 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 problems.Furthermore,the system matrix is very easy to assemble;thus,this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach.We conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al.(SIAM J.Numer.Anal.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 solution.The analysis applies to any consistent representation of the fnite element Hessian,and thus is applicable to the previous works making use of continuous Galerkin approximations.Numerical evidence is presented showing that the method works well also in a more general setting.展开更多
文摘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 tool.With this work,easier user interfaces based on Python and the unifed form language are provided to open Dune-Fem-DG for a broader audience.The Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs.
文摘We extend the fnite element method introduced by Lakkis and Pryer(SIAM J.Sci.Comput.33(2):786–801,2011)to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin(DG)framework.This is done by viewing the“fnite element Hessian”as an auxiliary variable in the formulation.Representing 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 problems.Furthermore,the system matrix is very easy to assemble;thus,this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach.We conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al.(SIAM J.Numer.Anal.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 solution.The analysis applies to any consistent representation of the fnite element Hessian,and thus is applicable to the previous works making use of continuous Galerkin approximations.Numerical evidence is presented showing that the method works well also in a more general setting.
