# Generalization of mixed multiscale finite element methods with applications

## Abstract

Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms. For practical purpose, we are interested in mixed finite element problems as they produce solutions with certain conservative properties. Existing multiscale methods for such problems include the mixed multiscale finite element methods. We show that for complicated problems, the mixed multiscale finite element methods may not be able to produce reliable approximations. This motivates the need of enrichment for coarse spaces. Two enrichment approaches are proposed, one is based on generalized multiscale finte element metthods (GMsFEM), while the other is based on spectral element-based algebraic multigrid (rAMGe). The former one, which is called mixed GMsFEM, is developed for both Darcy’s flow and linear elasticity. Application of the algorithm in two-phase flow simulations are demonstrated. For linear elasticity, the algorithm is subtly modified due to the symmetry requirement of the stress tensor. The latter enrichment approach is based on rAMGe. Themore »

- Authors:

- Texas A & M Univ., College Station, TX (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1260492

- Report Number(s):
- LLNL-TH-696037

- DOE Contract Number:
- AC52-07NA27344

- Resource Type:
- Thesis/Dissertation

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE

### Citation Formats

```
Lee, C S.
```*Generalization of mixed multiscale finite element methods with applications*. United States: N. p., 2016.
Web. doi:10.2172/1260492.

```
Lee, C S.
```*Generalization of mixed multiscale finite element methods with applications*. United States. doi:10.2172/1260492.

```
Lee, C S. Mon .
"Generalization of mixed multiscale finite element methods with applications". United States.
doi:10.2172/1260492. https://www.osti.gov/servlets/purl/1260492.
```

```
@article{osti_1260492,
```

title = {Generalization of mixed multiscale finite element methods with applications},

author = {Lee, C S},

abstractNote = {Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms. For practical purpose, we are interested in mixed finite element problems as they produce solutions with certain conservative properties. Existing multiscale methods for such problems include the mixed multiscale finite element methods. We show that for complicated problems, the mixed multiscale finite element methods may not be able to produce reliable approximations. This motivates the need of enrichment for coarse spaces. Two enrichment approaches are proposed, one is based on generalized multiscale finte element metthods (GMsFEM), while the other is based on spectral element-based algebraic multigrid (rAMGe). The former one, which is called mixed GMsFEM, is developed for both Darcy’s flow and linear elasticity. Application of the algorithm in two-phase flow simulations are demonstrated. For linear elasticity, the algorithm is subtly modified due to the symmetry requirement of the stress tensor. The latter enrichment approach is based on rAMGe. The algorithm differs from GMsFEM in that both of the velocity and pressure spaces are coarsened. Due the multigrid nature of the algorithm, recursive application is available, which results in an efficient multilevel construction of the coarse spaces. Stability, convergence analysis, and exhaustive numerical experiments are carried out to validate the proposed enrichment approaches. iii},

doi = {10.2172/1260492},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Mon Aug 01 00:00:00 EDT 2016},

month = {Mon Aug 01 00:00:00 EDT 2016}

}