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Two-dimensional (2-D) and three-dimensional (3-D) mesh-grading finite elements for problems with localized phenomena are presented. These quadrilaterals and hexahedrons permit mesh grading without element distortion and, if desired, can be used in a convenient recursive form. They are particularly well suited to adaptive finite element methods. Constraints associated with mesh grading are embedded in the basis functions, making the elements computationally efficient and easy to implement in standard finite element programs. Dramatic computational savings have been achieved. The computational complexity for an implicit analysis of a point load on a 3-D uniform mesh is $$O(n^ 7)$$. For the equivalent graded mesh it is $$O(\log_ 2n)$$. The corresponding 2-D analyses have computational complexities of $$O(n^ 4)$$ and $$O(\log_ 2n)$$.