Volume 15, Issue 3
Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations

Yalchin Efendiev, Juan Galvis, Guanglian Li & Michael Presho

Commun. Comput. Phys., 15 (2014), pp. 733-755.

Published online: 2014-03

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  • Abstract

In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], 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.

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@Article{CiCP-15-733, author = {}, title = {Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations}, journal = {Communications in Computational Physics}, year = {2014}, volume = {15}, number = {3}, pages = {733--755}, abstract = {

In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.020313.041013a}, url = {http://global-sci.org/intro/article_detail/cicp/7113.html} }
TY - JOUR T1 - Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations JO - Communications in Computational Physics VL - 3 SP - 733 EP - 755 PY - 2014 DA - 2014/03 SN - 15 DO - http://doi.org/10.4208/cicp.020313.041013a UR - https://global-sci.org/intro/article_detail/cicp/7113.html KW - AB -

In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], 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.

Yalchin Efendiev, Juan Galvis, Guanglian Li & Michael Presho. (2020). Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations. Communications in Computational Physics. 15 (3). 733-755. doi:10.4208/cicp.020313.041013a
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