Interpolation error bounds for curvilinear finite elements and their implications on adaptive mesh refinement
Moxey, D; Sastry, SP; Kirby, RM
Date: 10 August 2018
Journal
Journal of Scientific Computing
Publisher
Springer Verlag
Publisher DOI
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 ...
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.
Engineering
Faculty of Environment, Science and Economy
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