Singular Isoparametric Finite Elements

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Abstract
Curved isoparametric finite elements have proved very useful in many problems with complex geometries; they can however lead to serious numerical difficulties if the Jacobian of any of the local transformations vanishes within an element. Transformations are therefore invariably constrained to be small perturbations of a linear form, for which no such problems exist. This paper introduces a class of highly non-linear transformations for which the Jacobian vanishes at a single point on the element boundary, but which none the less are useful in approximating functions with non-removable singularities. Problems of this type arise from differential equations defined in regions with re-entrant corners. With this singular type of isoparametric approximation, it is not necessary to introduce special functions in the neighbourhood of the singularity.

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