This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.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.The first author acknowledges support from the EU Horizon 2020 project ExaFLOW(
grant 671571) and the PRISM project under EPSRC grant EP/L000407/1.
The work of the second author was supported in part by the NIH/NIGMS Center
for Integrative Biomedical Computing grant 2P41 RR0112553-12 and DOE NET
DE-EE0004449 grant. The work of the third author was supported in part by the DOE NET DE-EE0004449 grant and ARO W911NF1210375 (Program Manager:
Dr. Mike Coyle) grant
