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On a nonoverlapping additive S chwarz method for h ‐ p discontinuous G alerkin discretization of elliptic problems
Author(s) -
Krzyżanowski Piotr
Publication year - 2016
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22063
Subject(s) - mathematics , discretization , piecewise , discontinuous galerkin method , schwarz alternating method , degree of a polynomial , finite element method , domain decomposition methods , additive schwarz method , galerkin method , polynomial , degree (music) , partial differential equation , mathematical analysis , physics , thermodynamics , acoustics
The condition number of a discontinuous Galerkin h ‐ p finite element discretization preconditioned with a nonoverlapping additive Schwarz method is analyzed. We improve the result of Antonietti and Houston (J Sci Comput 46 (2011), 124–149), where a bound O ( p 2 H / h ) has been proved for a two‐level nonoverlapping additive Schwarz method with coarse problem using polynomials of degree p on a coarse mesh size H . In a more general framework, where the concurrency of the algorithm is increased by applying solvers on subdomains smaller than the coarse grid cells, we prove that the condition number of the preconditioned system is O ( p 2 / q ) · O ( H 2 / H h ) where q is the coarse space element degree polynomial and H ≤ H is the size of subdomain where local problems are solved in parallel. Our result also extends to the case of discontinuous coefficient, piecewise constant on the coarse grid, for a composite continuous–discontinuous Galerkin discretization. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1572–1590, 2016