Analytic element method

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 proscribed 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.


*Haitjema, H. M. (1995). Analytic element modeling of ground water flow, Academic Press, San Diego, CA.
*Strack, O. D. L. (1989). Groundwater Mechanics, Prentice Hall, Englewood Cliffs, NJ.

ee also

* Boundary element method

External links

* [ Analytic elements community wiki]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Boundary element method — The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form). It can be applied in many areas of engineering and …   Wikipedia

  • Spectral element method — In mathematics, the spectral element method is a high order finite element method.Introduced in a 1984 paper [A. T. Patera. A spectral element method for fluid dynamics Laminar flow in a channel expansion. Journal of Computational Physics ,… …   Wikipedia

  • Analytic — See also: Analysis Contents 1 Natural sciences 2 Philosophy 3 Social sciences …   Wikipedia

  • Method of lines — The method of lines (MOL, NMOL, NUMOL) (Schiesser, 1991; Hamdi, et al., 2007; Schiesser, 2009 ) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. MOL allows standard, general purpose… …   Wikipedia

  • Analytic Hierarchy Process — The Analytic Hierarchy Process (AHP) is a structured technique for helping people deal with complex decisions. Rather than prescribing a correct decision, the AHP helps people to determine one. Based on mathematics and human psychology, it was… …   Wikipedia

  • Element (mathematics) — In mathematics, an element or member of a set is any one of the distinct objects that make up that set. Contents 1 Sets 2 Notation and terminology 3 Cardinality of sets 4 Exampl …   Wikipedia

  • Spectral method — Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have… …   Wikipedia

  • Multigrid method — Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in (but not… …   Wikipedia

  • Discontinuous Galerkin method — Discontinuous Galerkin methods (DG methods) in mathematics form a class of numerical methods for solving partial differential equations. They combine features of the finite element and the finite volume framework and have been successfully… …   Wikipedia

  • Crank–Nicolson method — In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations.[1] It is a second order method in time, implicit in time, and is numerically …   Wikipedia