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

Mesh generation and adaptive refinement are largely driven by the objective of minimizing the bounds on the interpolation error of the solution of the partial differential equation (PDE) being solved. Thus, the characterization and analysis of interpolation error bounds for curved, high-order finite elements is often desired to efficiently obtain the solution of PDEs when using the finite element method (FEM). Although the order of convergence of the projection error in \(L^2\) is known for both straight-sided and curved elements (Botti in J Sci Comput 52(3):675–703, 2012), an \(L^{\infty }\) estimate as used when studying interpolation errors is not available. Using a Taylor series expansion approach, we derive an interpolation error bound for both straight-sided and curved high-order elements. The availability of this bound facilitates better node placement for minimizing interpolation error compared to the traditional approach of minimizing the Lebesgue constant as a proxy for interpolation error. This is useful for adaptation of the mesh in regions where increased resolution is needed and where the geometric curvature of the elements is high, e.g., boundary layer meshes. Our numerical experiments indicate that the error bounds derived using our technique are asymptotically similar to the actual error, i.e., if our interpolation error bound for an element is larger than it is for other elements, the actual error is also larger than it is for other elements. This type of bound not only provides an indicator for which curved elements to refine but also suggests whether one should use traditional h-refinement or should modify the mapping function used to define elemental curvature. We have validated our bounds through a series of numerical experiments on both straight-sided and curved elements, and we report a summary of these results.