A Staggered Discontinuous Galerkin Method for Quasi-Linear Second Order Elliptic Problems of Nonmonotone Type

Lina Zhao; Eun-Jae Park

doi:10.1515/cmam-2022-0081

Article

A Staggered Discontinuous Galerkin Method for Quasi-Linear Second Order Elliptic Problems of Nonmonotone Type

Lina Zhao and Eun-Jae Park

Published/Copyright: May 26, 2022

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From the journal Computational Methods in Applied Mathematics Volume 22 Issue 3

Abstract

In this paper, we present a priori and a posteriori analysis of a staggered discontinuous Galerkin (DG) method for quasi-linear second order elliptic problems of nonmonotone type. First, existence is proved by using Brouwer’s fixed point argument and uniqueness is verified utilizing Lipschitz continuity of the discrete solution map. Next, optimal a priori error estimates for both potential and flux variables are derived. Then the residual based a posteriori error estimates on the potential energy error and the flux $L 2$ error, respectively, are proposed. The flux error estimator makes use of a Helmholtz-type decomposition for the nonlinear system, which relies on appropriate choice of an auxiliary problem. While a priori error analysis is based on the observation that the staggered DG method can be viewed as a nonconforming approximation of the primal mixed formulation of the problem, a posteriori error estimation takes advantage of the primal formulation which can be obtained from the primal mixed formulation by eliminating the flux variable in the continuous setting. Finally, the theoretical findings are illustrated by numerical experiments.

Keywords: Staggered Discontinuous Galerkin Method; Quasi-Linear Elliptic Operator; A Posteriori Error Estimates; Linearization; Primal Formulation

MSC 2010: 65N15; 65N30; 65N50

Funding source: City University of Hong Kong

Award Identifier / Grant number: 7200699

Funding source: Ministry of Science and ICT, South Korea

Award Identifier / Grant number: NRF-2022R1A2B5B02002481

Funding statement: The first author was supported by a grant from City University of Hong Kong (Project No. 7200699). The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (NRF-2022R1A2B5B02002481).

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Received: 2022-04-01

Accepted: 2022-04-01

Published Online: 2022-05-26

Published in Print: 2022-07-01

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DOI: doi.org/10.1515/cmam-2022-0081

Keywords for this article

Staggered Discontinuous Galerkin Method; Quasi-Linear Elliptic Operator; A Posteriori Error Estimates; Linearization; Primal Formulation