On Interpolation Errors over Quadratic Nodal Triangular Finite Elements

Abstract

Summary. Interpolation techniques are used to estimate function values and their derivatives at those points for which a numerical solution of any equation is not explicitly evaluated. In particular, the shape functions are used to interpolate a solution (within an element) of a partial differential equation obtained by the finite element method. Mesh generation and quality improvement are often driven by the objective of minimizing the bounds on the error of the interpolated solution. For linear elements, the error bounds at a point have been derived as a composite function of the shape function values at the point and its distance from the element's nodes. We extend the derivation to quadratic triangular elements and visualize the bounds for both the function interpolant and the interpolant of its derivative. The maximum error bound for a function interpolant within an element is computed using the active set method for constrained optimization. For the interpolant of the derivative, we visually observe that the evaluation of the bound at the corner vertices is sufficient to find the maximum bound within an element. We characterize the bounds and develop a mesh quality improvement algorithm that optimizes the bounds through the movement (r-refinement) of both the corner vertices and edge nodes in a high-order mesh.

Interpolation Error Bounds for Curvilinear Finite Elements and Their Implications on Adaptive Mesh Refinement

“…As has been shown for linear [7] and quadratic [6] interpolation over a triangle, we will show that the following lemma holds at higher orders, thereby reducing terms in the equation above to zero.…”

“…In this section, we derive error bounds for the piecewise interpolation error of p-th order triangular elements with a q-th order reference-to-physical space mapping for a function and its derivative. The proofs for the error bounds closely follow the techniques employed by Johnson [7] and our previous paper [6].…”

“…Proof Just as it has been obtained by Johnson [7] and in our previous paper [6], the proof of the lemma is obtained by constructing appropriate polynomial functions w for every point in the domain K. To show Equation ( 6), let w(ξ) = c be a constant function. Then clearly w = πw, giving c = πw(ξ) = i λ i (ξ)w(ξ) = c i λ i (ξ) and…”

“…This again highlights that the error is bounded by coefficients of ξ i ξ j , where i+j > p. Furthermore, since we assume that v has sufficient regularity, we may take an explicit form of D p+1 so as to obtain a bound on the interpolation error as in [6], which leads to the following theorem:…”

“…In earlier work [6], we highlighted how a first-principles approach, using a Taylor series expansion of the function to be interpolated, leads to a pointwise bound on the interpolation error for second-order straight-sided elements. This bound, at a point in the high-order element, is proportional to the sum of the product of its distance from a high-order node of the triangle with the value of the corresponding shape function.…”