Two-scale sparse finite element approximations

Abstract

To reduce computational cost, we study some two-scale finite element approximations on sparse grids for elliptic partial differential equations of second order in a general setting. Over any tensor product domain $\Omega\subset\mathbb{R}^d$ with $d= 2,3$, we construct the two-scale finite element approximations for both boundary value and eigenvalue problems by using a Boolean sum of some existing finite element approximations on a coarse grid and some univariate fine grids and hence they are cheaper approximations. As applications, we obtain some new efficient finite element discretizations for the two classes of problem: The new two-scale finite element approximation on a sparse grid not only has the less degrees of freedom but also achieves a good accuracy of approximation.