The k-Version Finite Element Method for Singular Boundary-Value Problems with Application to Linear Fracture Mechanics
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This paper presents an application of the k -version finite element method to the numerical simulation of boundary value problems that contain singularity of the solution derivatives at certain point(s) in the domain. The theoretical solutions of such problems contain extremely isolated high solution gradients that approach infinity at the singular point(s); i.e., the solutions are not analytic at the singular point(s) but are analytic everywhere else. It is demonstrated that when numerical solutions of such problems are simulated in progressively increasing order scalar product spaces k , p ( e ), they approach the same characteristics in terms of differentiability as the theoretical solution as k is increased and in the limit k ? 8 , the numerical solutions have exactly the same global differentiability characteristics as the theoretical solutions. A two-dimensional linear elastic fracture mechanics problem is used as a model problem to illustrate the salient features of the k -version finite element method.