An official website of the United States government
We present a finite element technique for approximating the surface Hessian of a discrete scalar function on triangulated surfaces embedded in $$\R^{3}$$, with or without boundary. We then extend the method to compute approximations of the full shape operator of the underlying surface using only the known discrete surface. The method is based on the Hellan--Herrmann--Johnson (HHJ) element and does not require any ad-hoc modifications. Convergence is established provided the discrete surface satisfies a Lagrange interpolation property related to the exact surface. The convergence rate, in $L^2$, for the shape operator approximation is $O(h^m)$, where $$m \geq 1$$ is the polynomial degree of the surface, i.e. the method converges even for piecewise linear surface triangulations. For surfaces with boundary, some additional boundary data is needed to establish optimal convergence, e.g. boundary information about the surface normal vector or the curvature in the co-normal direction. Numerical examples are given on non-trivial surfaces that demonstrate our error estimates and the efficacy of the method.
more »
« less
Convergence of Lagrange finite elements for the Maxwell eigenvalue problem in two dimensions
Abstract We consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell–Sabin triangulations, piecewise quadratic elements on Clough–Tocher triangulations and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eigenvalue approximations are very sensitive to nearly singular vertices, i.e., vertices that fall on exactly two ‘almost’ straight lines.
more »
« less
- Award ID(s):
- 2011733
- PAR ID:
- 10432854
- Date Published:
- Journal Name:
- IMA Journal of Numerical Analysis
- Volume:
- 43
- Issue:
- 2
- ISSN:
- 0272-4979
- Page Range / eLocation ID:
- 663 to 691
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples.more » « less
-
New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order r+1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in Hp for any p≥1. We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used.more » « less
-
We propose an algorithm to solve convex and concave fractional programs and their stochastic counterparts in a common framework. Our approach is based on a novel reformulation that involves differences of square terms in the constraints, and subsequent employment of piecewise-linear approximations of the concave terms. Using the branch-and-bound (B\&B) framework, our algorithm adaptively refines the piecewise-linear approximations and iteratively solves convex approximation problems. The convergence analysis provides a bound on the optimality gap as a function of approximation errors. Based on this bound, we prove that the proposed B\&B algorithm terminates in a finite number of iterations and the worst-case bound to obtain an $$\epsilon$$-optimal solution reciprocally depends on the square root of $$\epsilon$$. Numerical experiments on Cobb-Douglas production efficiency and equitable resource allocation problems support that the algorithm efficiently finds a highly accurate solution while significantly outperforming the benchmark algorithms for all the small size problem instances solved. A modified branching strategy that takes the advantage of non-linearity in convex functions further improves the performance. Results are also discussed when solving a dual reformulation and using a cutting surface algorithm to solve distributionally robust counterpart of the Cobb-Douglas example models.more » « less
-
Abstract We establish several useful estimates for a non-conforming 2-norm posed on piecewise linear surface triangulations with boundary, with the main result being a Poincaré inequality.We also obtain equivalence of the non-conforming 2-norm posed on the true surface with the norm posed on a piecewise linear approximation.Moreover, we allow for free boundary conditions.The true surface is assumed to be C 2 , 1 C^{2,1} when free conditions are present; otherwise, C 2 C^{2} is sufficient.The framework uses tools from differential geometry and the closest point map (see [G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations , Lecture Notes in Math. 1357, Springer, Berlin (1988), 142–155]) for approximating the full surface Hessian operator.We also present a novel way of applying the closest point map when dealing with surfaces with boundary.Connections with surface finite element methods for fourth-order problems are also noted.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imanum/drab104
