The generating analytic element approach with application to the modified Helmholtz equation

Abstract

In this paper a new method for obtaining functions with a given singular behavior that satisfy a class of partial differential equations is presented. Differential equations of this class contain operators of the form ${\nabla^{2n}}$, where n is a positive integer. The method uses Wirtinger calculus which enables one to invert the Laplacian in combination with the decomposition method introduced by Adomian at the end of the twentieth century. The procedure uses a singular holomorphic function as its basis, and constructs the solution term by term as an infinite series of functions; the process consists of an infinite number of steps of integration. This method is applied to construct a number of singular solutions to the modified Helmholtz equation in the context of groundwater flow. These functions are discharge potentials, which are two-dimensional functions by definition. The gradient of the discharge potential is the vertically integrated flow over the thickness of an aquifer, or water-bearing layer. The discharge potentials of interest here are those used in the analytic element method. This method, as originally conceived, relies on the superposition of suitably chosen holomorphic functions, and is a form of a method known as the Trefftz method, not to be confused with the Trefftz method applied to finite element techniques. The main analytic elements used are singular line elements, characterized by either a jump along the element in the tangential or the normal component of the discharge vector. The analytic line elements for the case of divergence-free irrotational flow are well established and many of these are forms of singular Cauchy integrals. Application of the analytic element method to more general cases of flow, governed for example by the modified Helmholtz equation (flow in systems of aquifers separated by leaky layers) and the heat equation (transient flow) is possible using the method presented in this paper. The latter application is beyond the scope of this paper, but it is worth noting that for that case the constant that occurs in the modified Helmholtz is replaced by a general function of time and application of Laplace transforms can be avoided. A method for constructing such functions is presented; the procedure for constructing these functions is referred to as the generating analytic element approach. Application of this approach requires the existence of the holomorphic singular line element. The approach is discussed and an example for the case of a line-sink for a system of two aquifers separated by a leaky layer and bounded above by in impermeable boundary is presented.