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Polynomial basis functions are the ubiquitous workhorse of high-order finite element methods, but their generality comes at a price of high computational cost and fragility in the face of underresolution. In this paper, a method is presented for constructing a posteriori tailored, generally nonpolynomial, basis functions for approximating a solution and computing outputs of a system of equations. This method is similar to solution-based adaptation, in which elements of the computational mesh are sized and oriented based on characteristics of the solution. The method takes advantage of existing infrastructure in high-order methods: the reference-to-global mapping used in constructing curved elements. By optimizing this mapping, elements are warped to make them ideally suited for representing a target solution or computing a scalar output from the solution. Guidelines on generating a good initial guess and choosing a generalized set of optimization parameters are provided to minimize tuning time and to intr...

Improving High-Order Finite Element Approximation Through Geometrical Warping