Using p-Refinement to Increase Boundary Derivative Convergence Rates
This addresses the need for accurate boundary derivative computation in fluid-structure interaction and conjugate heat transfer, but the method is demonstrated only in simple settings and is incremental.
The paper proposes a novel continuous finite element method using p-refinement near boundaries to increase the accuracy of boundary derivative approximations for elliptic problems, achieving higher convergence rates verified numerically in 1D and 2D.
Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating accurate approximations of boundary derivatives of elliptic problems: instead of postprocessing, we describe a new continuous finite element method based on p-refinement of cells adjacent to the boundary to increase the approximation order of the derivative on the boundary itself. We prove that the order of the approximation on the p-refined cells is, in 1D, determined by the rate of convergence at the knot connecting the higher and lower order cells and that this idea can be extended, in some simple settings, to 2D problems. We verify this rate of convergence numerically with a series of experiments in both 1D and 2D.