@Article{CiCP-14-4,
author = {},
title = {Multiscale Finite Element Methods for Flows on Rough Surfaces},
journal = {Communications in Computational Physics},
year = {2013},
volume = {14},
number = {4},
pages = {979--1000},
abstract = {<p style="text-align: justify;">In this paper, we present the Multiscale Finite Element Method (MsFEM)
for problems on rough heterogeneous surfaces. We consider the diffusion equation on
oscillatory surfaces. Our objective is to represent small-scale features of the solution
via multiscale basis functions described on a coarse grid. This problem arises in many
applications where processes occur on surfaces or thin layers. We present a unified
multiscale finite element framework that entails the use of transformations that map
the reference surface to the deformed surface. The main ingredients of MsFEM are (1)
the construction of multiscale basis functions and (2) a global coupling of these basis
functions. For the construction of multiscale basis functions, our approach uses the
transformation of the reference surface to a deformed surface. On the deformed surface, multiscale basis functions are defined where reduced (1D) problems are solved
along the edges of coarse-grid blocks to calculate nodal multiscale basis functions. Furthermore, these basis functions are transformed back to the reference configuration.
We discuss the use of appropriate transformation operators that improve the accuracy
of the method. The method has an optimal convergence if the transformed surface
is smooth and the image of the coarse partition in the reference configuration forms
a quasiuniform partition. In this paper, we consider such transformations based on
harmonic coordinates (following H. Owhadi and L. Zhang [Comm. Pure and Applied
Math., LX(2007), pp. 675–723]) and discuss gridding issues in the reference configuration. Numerical results are presented where we compare the MsFEM when two
types of deformations are used for multiscale basis construction. The first deformation
employs local information and the second deformation employs a global information.
Our numerical results show that one can improve the accuracy of the simulations when
a global information is used.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.170512.310113a},
url = {https://global-sci.com/article/80586/multiscale-finite-element-methods-for-flows-on-rough-surfaces}
}