Analytic Element Method

The analytic element method (AEM) is a numerical method used for the solution of partial differential equations. It was initially developed by O.D.L. Strack at the University of Minnesota. It is similar in nature to the boundary element method (BEM), as it does not rely upon discretization of volumes or areas in the modeled system; only internal and external boundaries are discretized. One of the primary distinctions between AEM and BEMs is that the boundary integrals are calculated analytically.

The analytic element method is most often applied to problems of groundwater flow governed by the Poisson equation, though it is applicable to a variety of linear partial differential equations, including the Laplace, Helmholtz, and biharmonic equations.

The basic premise of the analytic element method is that, for linear differential equations, elementary solutions may be superimposed to obtain more complex solutions. A suite of 2D and 3D analytic solutions ("elements") are available for different governing equations. These elements typically correspond to a discontinuity in the dependent variable or its gradient along a geometric boundary (e.g., point, line, ellipse, circle, sphere, etc.). This discontinuity has a specific functional form (usually a polynomial in 2D) and may be manipulated to satisfy Dirichlet, Neumann, or Robin (mixed) boundary conditions. Each analytic solution is infinite in space and/or time. In addition, each analytic solution contains degrees of freedom (coefficients) that may be calculated to meet prescribed boundary conditions along the element's border. To obtain a global solution (i.e., the correct element coefficients), a system of equations is solved such that the boundary conditions are satisfied along all of the elements (using collocation, least-squares minimization, or a similar approach). Notably, the global solution provides a spatially continuous description of the dependent variable everywhere in the infinite domain, and the governing equation is satisfied everywhere exactly except along the border of the element, where the governing equation is not strictly applicable due to the discontinuity.