Equivalence between the DPG method and the exponential integrators for linear parabolic problems

JOURNAL OF COMPUTATIONAL PHYSICS

卷:429

Article

Munoz-Matute, Judit¹  Pardo, David^1,2,3  Demkowicz, Leszek⁴ 

[1] Basque Ctr Appl Math, Bilbao, Bcam, Spain

[2] Univ Basque Country UPV EHU, Leioa, Spain

[3] Basque Fdn Sci, IKERBASQUE, Bilbao, Spain

[4] Univ Texas Austin, Oden Inst Computat Engn & Sci, Austin, TX USA

关键词: DPG method;    Ultraweak formulation;    Optimal test functions;    Exponential integrators;    Linear parabolic problems;    ODE systems;   

DOI  :  10.1016/j.jcp.2020.110016

来源: Elsevier

【 摘 要 】

The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well established numerical methods for solving Partial Differential Equations (PDEs) and stiff systems of Ordinary Differential Equations (ODEs), respectively. In this work, we apply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general first order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D + time linear parabolic PDEs after discretizing in space by the finite element method. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free