Interpolation error bounds for curvilinear finite elements and their implications on adaptive mesh refinement

Abstract

Mesh generation and adaptive renement are largely driven by the objective of minimizing the bounds on the interpolation error of the solution of the partial di erential equation (PDE) being solved. Thus, the characterization and analysis of interpolation error bounds for curved, high-order nite elements is often desired to e ciently obtain the solution of PDEs when using the nite element method (FEM). Although the order of convergence of the projection error in L2 is known for both straight-sided and curved-elements [1], an L1 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 re ne but also suggests whether one should use traditional h-re nement or should modify the mapping function used to de ne 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.