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Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains
Author(s) -
Pramanick Tamal,
Sinha Rajen Kumar
Publication year - 2018
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.22293
Subject(s) - mathematics , finite element method , discretization , mathematical analysis , piecewise , nonlinear system , norm (philosophy) , backward euler method , piecewise linear function , domain (mathematical analysis) , regular grid , grid , boundary value problem , boundary (topology) , geometry , physics , quantum mechanics , political science , law , thermodynamics
We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math ., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L ∞ ( L 2 ) norm. Numerical experiment is carried out for an L ‐shaped domain to illustrate our theoretical findings.