A ROBUST DPG METHOD FOR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS

被引：28

作者：

Heuer, Norbert ^{[1
]}

Karkulik, Michael ^{[2
]}

机构：

[1] Pontificia Univ Catolica Chile, Fac Matemat, Santiago, Chile

[2] Univ Tecn Federico Santa Maria, Dept Matemat, Valparaiso, Chile

来源：

SIAM JOURNAL ON NUMERICAL ANALYSIS | 2017年 / 55卷 / 03期

基金：

美国国家科学基金会;

关键词：

reaction-dominated diffusion; singularly perturbed problem; boundary layers; discontinuous Petrov-Galerkin method; PETROV-GALERKIN DISCRETIZATION; FINITE-ELEMENT METHODS; OPTIMAL TEST SPACE; BALANCED NORMS; CONVERGENCE; FORMULATION;

D O I：

10.1137/15M1041304

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We present and analyze a discontinuous Petrov-Galerkin method with optimal test functions for a reaction-dominated diffusion problem in two and three space dimensions. We start with an ultraweak formulation that comprises parameters alpha, beta to allow for general epsilon-dependent weightings of three field variables (epsilon being the small diffusion parameter). Specific values of alpha and beta imply robustness of the method, that is, a quasi-optimal error estimate with a constant that is independent of epsilon. Moreover, these values lead to a norm for the field variables that is known to be balanced in epsilon for model problems with typical boundary layers. Several numerical examples underline our theoretical estimates and reveal stability of approximations even for very small epsilon.

引用

页码：1218 / 1242

页数：25

共 23 条

[1] Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem [J].

Apel, T ;

Lube, G .

APPLIED NUMERICAL MATHEMATICS, 1998, 26 (04) :415-433

[2]

Bansch E., 1991, Impact of Computing in Science and Engineering, V3, P181, DOI 10.1016/0899-8248(91)90006-G

[3] A Petrov-Galerkin discretization with optimal test space of a mild-weak formulation of convection-diffusion equations in mixed form [J].

Broersen, Dirk ;

Stevenson, Rob P. .

IMA JOURNAL OF NUMERICAL ANALYSIS, 2015, 35 (01) :39-73

[4] A robust Petrov-Galerkin discretisation of convection-diffusion equations [J].

Broersen, Dirk ;

Stevenson, Rob .

COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2014, 68 (11) :1605-1618

[5] A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms [J].

Chan, Jesse ;

Heuer, Norbert ;

Tan Bui-Thanh ;

Demkowicz, Leszek .

COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2014, 67 (04) :771-795

[6] ANALYSIS OF THE DPG METHOD FOR THE POISSON EQUATION [J].

Demkowicz, L. ;

Gopalakrishnan, J. .

SIAM JOURNAL ON NUMERICAL ANALYSIS, 2011, 49 (05) :1788-1809

[7] A Class of Discontinuous Petrov-Galerkin Methods. II. Optimal Test Functions [J].

Demkowicz, L. ;

Gopalakrishnan, J. .

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, 2011, 27 (01) :70-105

[8]

Demkowicz L., 2014, 1436 ICES U TEX AUST

[9] ROBUST DPG METHOD FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS [J].

Demkowicz, Leszek ;

Heuer, Norbert .

SIAM JOURNAL ON NUMERICAL ANALYSIS, 2013, 51 (05) :2514-2537

[10] A class of discontinuous Petrov-Galerkin methods. Part III: Adaptivity [J].

Demkowicz, Leszek ;

Gopalakrishnan, Jay ;

Niemi, Antti H. .

APPLIED NUMERICAL MATHEMATICS, 2012, 62 (04) :396-427

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