Solution of a Convolution Equation with Unbounded Domain via Multigrid

Steven M. McKay

Dept. of Mathematics Brigham Young University 310 TMCB Provo, Utah


The solution of integro-differential equations involving a convolution are of interest to researchers in the material science fields due to the similarity in solution to models such as Allen-Cahn which describe behavior of mixed alloy metals during the quenching process. It is well known that current models for this process produce regions which have a high concentration of one metal. We consider a convolution equation which is a continuous analogue to the discrete models used to study this process. The movement of the alloy from a mixed or homogeneous state to a mixed state appears to occur on two levels. There is a rapid change phase which develops the regions of high concentration, and a slower phase which occurs after the regions are developed. This slow phase is mainly concerned with sharpening of the interface between two different regions. We will apply a multigrid algorithm to the equation in two different ways: First, we will apply multigrid on an irregular grid in order to more efficiently solve the problem. Second, we will use irregular time stepping on the grids to accelerate the convergence of the solution to steady state.