Theory and practice of the measured equation of invariance

The measured equation of invariance (MEI) is a simple technique used to derive finite difference type local equations at mesh boundaries, where the conventional finite difference approach fails [1,2]. The equation is derived based on the fact that it is invariant to the incident fields. Finite difference and finite element equations are examples of such local linear equations satisfying such invariance, but they are derived from the conventional wave equations. MEI is an example of alternatives to the wave-equation-based local discrete equations. Conventionally, finite difference or finite element meshes span from boundary to boundary, or to any surface where an absorbing boundary condition can be simulated. It was demonstrated that the MEI technique can be used to terminate meshes very close to the object boundary and still strictly preserves the sparsity of the finite difference equations. It results in dramatic savings in computing time and memory needs. In this paper, the basic theory of the MEI method will be presented. We shall show that MEI can be readily applied to boundary value problems of elliptical partial differential equations, such as, Laplace equations, wave equations and Maxwell´s equations resulting in great savings in both storage and cpu time over the conventional methods.