Generalized multiscale finite element methods. nonlinear elliptic equations
KAUST DepartmentNumerical Porous Media SRI Center (NumPor)
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
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AbstractIn this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in , in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples. © 2014 Global-Science Press.
SponsorsY. Efendiev's work is partially supported by the DOE and NSF (DMS 0934837 and DMS 0811180). J. Galvis would like to acknowledge partial support from DOE. This publication is based in part on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
PublisherGlobal Science Press