Abstract
We derive error estimates for the piecewise linear finite element approximation of the Laplace–Beltrami operator on a bounded, orientable, \(C^3\), surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.
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Johnny Guzman and Marcus Sarkis are supported by National Science Foundation (Grant Nos. DMS #1620100 and NSF-MPS 1522663).
Appendix A: One Technical Result
Appendix A: One Technical Result
Lemma 19
Assume that \(\Gamma \) is a \(C^3\) two-dimensional compact orientable surface without boundary, and that \(u\in H^2(\Gamma )\). Then
Proof
In what follows, we use the two identities [17]
and
for all \(C^3(\Gamma )\) functions u, v. Also we will use that of course
We assume for the proof that \(u\in C^3(\Gamma )\), and the general result follows from density arguments. Following [17, Lemma 3.2], and using the Einstein summation convention,
To handle the first term on the right hand side, we use (53) and the fact that \(\underline{D}_{ii} u= \Delta _{\Gamma } u\) to write:
where we used (54) in the last equation. But \(\underline{D}_{ki}u\nu _i=\underline{D}_k(\underline{D}_iu\nu _i)-\underline{D}_iuH_{ki}=-\underline{D}_iuH_{ki}\), and then
In the last equation we used (52) and (54). This completes the proof. \(\square \)
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Guzman, J., Madureira, A., Sarkis, M. et al. Analysis of the Finite Element Method for the Laplace–Beltrami Equation on Surfaces with Regions of High Curvature Using Graded Meshes. J Sci Comput 77, 1736–1761 (2018). https://doi.org/10.1007/s10915-017-0580-y
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DOI: https://doi.org/10.1007/s10915-017-0580-y