Computation of eigenvalues by numerical upscaling
This provides a computationally efficient method for solving multi-scale eigenvalue problems in reservoir modeling and material science.
The authors develop numerical upscaling techniques that compute lowermost eigenvalues of elliptic PDEs with superconvergent accuracy using a low-dimensional generalized finite element space, with rigorous error bounds.
We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we consider benchmark multi-scale eigenvalue problems in reservoir modeling and material science. We compute a low-dimensional generalized (possibly mesh free) finite element space that preserves the lowermost eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of $H^1_0(Ω)$ by means of a certain Clément-type quasi-interpolation operator.