scholarly journals Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations

2014 ◽  
Vol 15 (3) ◽  
pp. 733-755 ◽  
Author(s):  
Yalchin Efendiev ◽  
Juan Galvis ◽  
Guanglian Li ◽  
Michael Presho
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Problem Formulation ◽  
Nonlinear Elliptic ◽  
Continuous Galerkin ◽  
Previous Solution ◽  
Multiscale Finite Element ◽  
Multiscale Finite Element Methods

AbstractIn 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.

scholarly journals A hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac delta source

2022 ◽  
Author(s):  
Haitao Leng ◽  
Yanping Chen
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Elliptic Equations ◽  
A Posteriori Error Estimates ◽  
Second Order ◽  
Dirac Delta ◽  
Hybridizable Discontinuous Galerkin ◽  
Hybridizable Discontinuous Galerkin Method ◽  
Second Order Elliptic Equations

In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in $L^2$-norm is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in $L^2$-norm and $W^{1,p}$-seminorm are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.

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