High-dimensional finite element method for multiscale linear elasticity
Bingxing Xia, Viêt Hà Hòang · IMA Journal of Numerical Analysis · 2014
Sparse tensor product finite elements (FEs) are used for discretizing the high-dimensional multiscale homogenized equation of a linear elasticity equation in ℝd that depends on n microscopic scales. The solution to the homogenized equation that describes the macroscopic property and the corrector terms that encode the microscopic information are obtained with accuracy and complexity essentially equal to that for solving a one-scale macroscopic equation in ℝd. An approximation for the solution of the multiscale equation in terms of the FE solution of this high-dimensional problem is obtained. For two-scale problems, an explicit error for this approximation in terms of the FE mesh width and the microscopic scale is deduced. Numerical examples of two- and three-scale elasticity equations in two dimensions confirm the analysis.