Dedner, Andreas and Pryer, Tristan (2022) Discontinuous Galerkin methods for a class of nonvariational problems. Communications on Applied Mathematics and Computation, 4 . pp. 634-656. doi:10.1007/s42967-021-00133-6 ISSN 2096-6385.

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Abstract

We extend the finite element method introduced by Lakkis and Pryer [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 ``finite element Hessian'' as an auxiliary variable in the formulation. Representing the finite 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 unified framework set out in Arnold et. al. [2001]. 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 finite element Hessian, thus is applicable to the previous works making use of continuous Galerkin approximations. Numerical evidence is the presented showing that the method works well also in a more general setting.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Differential equations, Partial, Numerical analysis, Galerkin methods, Finite element method
Journal or Publication Title:	Communications on Applied Mathematics and Computation
Publisher:	Springer
ISSN:	2096-6385
Official Date:	June 2022
Dates:	Date Event June 2022 Published 13 September 2021 Available 22 March 2021 Accepted
Volume:	4
Page Range:	pp. 634-656
DOI:	10.1007/s42967-021-00133-6
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	23 March 2021
Date of first compliant Open Access:	29 September 2021
Related URLs:	Publisher

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