Neumann-Neumann methods for a DG discretization on geometrically nonconforming substructures

A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming....

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Vydáno v:Numerical methods for partial differential equations Ročník 28; číslo 4; s. 1194 - 1226
Hlavní autoři: Dryja, Maksymilian, Galvis, Juan, Sarkis, Marcus
Médium: Journal Article
Jazyk:angličtina
Vydáno: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.07.2012
Témata:
discontinuous Galerkin method
elliptic problems with discontinuous coefficients
finite element method
interior penalty discretization
Neumann-Neumann algorithms
nonconforming decomposition
preconditioners
Schwarz methods
ISSN:0749-159X, 1098-2426
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Shrnutí:A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ωi, a conforming finite element space associated to a triangulation \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} {\mathcal{T}}_{h_i}(\Omega_i)\end{align*}\end{document} is introduced. To handle the nonmatching meshes across ∂Ωi, a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann‐Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across ∂Ωi, a condition number estimate \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} C(1 + \max_i\log \frac{H_i}{h_i})^2\end{align*}\end{document} is established with C independent of hi, Hi, hi/hj, and the jumps of the coefficients. The method is well suited for parallel computations and can be straightforwardly extended to three dimensional problems. Numerical results are included. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012
Bibliografie:ark:/67375/WNG-KSJ13VDG-S
ArticleID:NUM20678
istex:2A9695517EBE0016CABE667A37EEF642DB860B68
Polish Sciences Foundation - No. NN201006933
PEC-PG-CAPES/Brazil
CNPq/Brazil - No. 300964/2006-4
ISSN:0749-159X
1098-2426
DOI:10.1002/num.20678