Limitless Analytic Elements

• Published in: Water resources research 2018-02, Vol.54 (2), p.1174-1190
• Main Author: Strack, O. D. L.
• Format: Article
• Language: English
• Subjects: analytic element method; Boundaries; complex variables; groundwater flow; Hydraulic conductivity; Iterative methods; iterative solution; line elements; Mathematical analysis; Mathematical models; Modelling; Rivers; Strings; superposition
• ISSN: 0043-1397; 1944-7973
• DOI: 10.1002/2017WR022117

We present equations for new limitless analytic line elements. These elements possess a virtually unlimited number of degrees of freedom. We apply these new limitless analytic elements to head‐specified boundaries and to problems with inhomogeneities in hydraulic conductivity. Applications of these new analytic elements to practical problems involving head‐specified boundaries require the solution of a very large number of equations. To make the new elements useful in practice, an efficient iterative scheme is required. We present an improved version of the scheme presented by Bandilla et al. (2007), based on the application of Cauchy integrals. The limitless analytic elements are useful when modeling strings of elements, rivers for example, where local conditions are difficult to model, e.g., when a well is close to a river. The solution of such problems is facilitated by increasing the order of the elements to obtain a good solution. This makes it unnecessary to resort to dividing the element in question into many smaller elements to obtain a satisfactory solution. Key Points Presentation of new very high‐order two‐dimensional analytic elements, effectively limitless, bounded only in accuracy by machine precision Demonstration that limitless analytic elements can be strung together without loss of accuracy; they closely reproduce singularities Presentation of an efficient and accurate method for determining discharges of line‐sinks in an iterative scheme