The Finite Line Integration Method (FLIM) - A Fast Variant of Finite Element Modelling

A new way of modelling and solving partial differential equations is derived. While the finite element (FE) methodology is based on the integration over finite elements, this novel approach utilizes edges (finite lines) instead, which results in a finite line integration (FLIM). We present how the FE-computations over elements (areas or volumes) can be rewritten for the stiffness matrix based on edges and how the load can be obtained along these finite lines. It is shown that the performance of the new method can be made identical to the performance of the finite element method (FEM) for linear triangular or tetrahedral elements independent of the mesh being structured or unstructured. The new method, however, requires much lower storage than FEM, especially for three-dimensional problems, but yields the same approximation error and convergence rate as the finite element method. Further, this approach allows for a direct computation of stiffness matrices at edge level, which can advantageously be used for the development of efficient linear iterative solvers.