1,119 research outputs found
Parallel High-Order Anisotropic Meshing Using Discrete Metric Tensors
This paper presents a metric-aligned meshing algorithm that relies on the Lp-Centroidal Voronoi Tesselation approach. A prototype of this algorithm was first presented at the Scitech conference of 2018 and this work is an extension to that paper. At the end of the previously presented work, a set of problems were mentioned which we are trying to address in this paper. First, we show a significant improvement in code performance since we were limited to present relatively benign (analytical) test cases. Second, we demonstrate here that we are able to rely on discrete metric data that is delivered by a Computational Fluid Dynamics (CFD) solver. Third, we demonstrate how to generate high-order curved elements that are aligned with the underlying discrete metric field
Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant
The purpose of this paper is to establish that for any compact, connected
C^{\infty} Riemannian manifold there exists a robust family of kernels of
increasing smoothness that are well suited for interpolation. They generate
Lagrange functions that are uniformly bounded and decay away from their center
at an exponential rate. An immediate corollary is that the corresponding
Lebesgue constant will be uniformly bounded with a constant whose only
dependence on the set of data sites is reflected in the mesh ratio, which
measures the uniformity of the data.
The analysis needed for these results was inspired by some fundamental work
of Matveev where the Sobolev decay of Lagrange functions associated with
certain kernels on \Omega \subset R^d was obtained. With a bit more work, one
establishes the following: Lebesgue constants associated with surface splines
and Sobolev splines are uniformly bounded on R^d provided the data sites \Xi
are quasi-uniformly distributed. The non-Euclidean case is more involved as the
geometry of the underlying surface comes into play. In addition to establishing
bounded Lebesgue constants in this setting, a "zeros lemma" for compact
Riemannian manifolds is established.Comment: 33 pages, 2 figures, new title, accepted for publication in SIAM J.
on Math. Ana
Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries
We construct asymptotically Euclidean solutions of the vacuum Einstein
constraint equations with an apparent horizon boundary condition. Specifically,
we give sufficient conditions for the constant mean curvature conformal method
to generate such solutions. The method of proof is based on the barrier method
used by Isenberg for compact manifolds without boundary, suitably extended to
accommodate semilinear boundary conditions and low regularity metrics. As a
consequence of our results for manifolds with boundary, we also obtain
improvements to the theory of the constraint equations on asymptotically
Euclidean manifolds without boundary.Comment: 27 pages, 1 figure, TeX, v3. Final version to appear in CMP.
Exposition has been extensively tightened and the proof of Proposition 3.5
has been simplifie
How a nonconvergent recovered Hessian works in mesh adaptation
Hessian recovery has been commonly used in mesh adaptation for obtaining the
required magnitude and direction information of the solution error.
Unfortunately, a recovered Hessian from a linear finite element approximation
is nonconvergent in general as the mesh is refined. It has been observed
numerically that adaptive meshes based on such a nonconvergent recovered
Hessian can nevertheless lead to an optimal error in the finite element
approximation. This also explains why Hessian recovery is still widely used
despite its nonconvergence. In this paper we develop an error bound for the
linear finite element solution of a general boundary value problem under a mild
assumption on the closeness of the recovered Hessian to the exact one.
Numerical results show that this closeness assumption is satisfied by the
recovered Hessian obtained with commonly used Hessian recovery methods.
Moreover, it is shown that the finite element error changes gradually with the
closeness of the recovered Hessian. This provides an explanation on how a
nonconvergent recovered Hessian works in mesh adaptation.Comment: Revised (improved proofs and a better example
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