A family of exponentially-gradient elements for numerical computation of singular boundary value problems

In this paper, a new family of single-parameter exponentially gradient elements (EG-elements) are introduced, which can be used in various numerical procedures such as boundary and finite element methods. These elements have the ability to accurately interpolate the unknown values in regions, where either high gradient or singularity of the unknown field occurs. The shape functions of two-dimensional EG-elements are high gradient at either a corner of the element or at an edge of the element. Another advantage of this element is that the regular quadratic shape functions are obtained as a special case of EG-elements by adjusting the single parameter of the element, which allows this element to be used as regular Lagrange quadratic element, where it is appropriate. Some mixed boundary value problems are solved with the use of EG-elements in a boundary element program to show the capability of these elements for capturing the solution with less number of elements and higher accuracy.