Finite Element Methods with Coordinate Charts for Solving Elliptic Equations on Manifolds

We apply finite element methods to elliptic problems on compact Riemannian manifolds. The elliptic equation on manifolds is reduced to coupled equations on Euclidean spaces by coordinate charts. The advantage of this strategy is to avoid global triangulations on curved manifolds in the finite element method. The resulting finite element problem can be solved globally, or locally by nonoverlapping or overlapping domain decomposition methods. The method is illustrated by a 4D problem. We present the theory and computational results. AMS subject classifications. Primary 65N30; Secondary 46E25, 20C20