Efficient solution of lattice equations by the recovery method Part I: Scalar elliptic problems

In this paper, an efficient solver for high dimensional lattice equations will be introduced. We will present a new concept, the recovery method, to define a bilinear form on the continuous level which has equivalent energy as the original lattice equation. The finite element discretisation of the continuous bilinear form will lead to a stiffness matrix which serves as an quasi-optimal preconditioner for the lattice equations. Since a large variety of efficient solvers are available for linear finite element problems the new recovery method allows to apply these solvers for unstructured lattice problems.