Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1402.6001

A study on anisotropic mesh adaptation for finite element approximation of eigenvalue problems with anisotropic diffusion operators

Jingyue Wang·Weizhang Huang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Anisotropic mesh adaptation is studied for the linear finite element solution of eigenvalue problems with anisotropic diffusion operators. The M-uniform mesh approach is employed with which any nonuniform mesh is characterized mathematically as a uniform one in the metric specified by a metric tensor. Bounds on the error in the computed eigenvalues are established for quasi-M-uniform meshes. Numerical examples arising from the Laplace-Beltrami operator on parameterized surfaces, nonlinear diffusion, thermal diffusion in a magnetic field in plasma physics, and the Laplacian operator on an L-shape domain are presented. Numerical results show that anisotropic adaptive meshes can lead to more accurate computed eigenvalues than uniform or isotropic adaptive meshes. They also confirm the second order convergence of the error that is predicted by the theoretical analysis. The effects of approximation of curved boundaries on the computation of eigenvalue problems is also studied in two dimensions. It is shown that the initial mesh used to define the geometry of the physical domain should contain at least \sqrt{N} boundary points to keep the effects of boundary approximation at the level of the error of the finite element approximation, where N is the number of the elements in the final adaptive mesh. Only about N^{1/3} boundary points in the initial mesh are needed for boundary value problems. This implies that the computation of eigenvalue problems is more sensitive to the boundary approximation than that of boundary value problems.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jingyue Wang, Weizhang Huang. 2014-02-24. A study on anisotropic mesh adaptation for finite element approximation of eigenvalue problems with anisotropic diffusion operators. https://doi.org/10.1137/140958554

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

How many continuous measurements are needed to learn a vector?

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present applications to infinite-dimensional approximation problems.

math.NA↗
arXiv2026

IterativeCUR: Large Rank-Adaptive Approximation From a Small Recycled Sketch

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or Randomized SVD. Although these SVD approaches---especially Randomized SVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work. IterativeCUR is adaptive in the sense that it takes as an input parameter the desired tolerance $ε$ and outputs (with arbitrarily high probability) an approximation of error bounded by $ε$, rather than requiring an a priori guess of the numerical rank. IterativeCUR typically runs significantly faster than both existing CUR algorithms and techniques such as Randomized SVD. Its asymptotic complexity is $\mathcal{O}(mn + (m+n)r^2)$ for an $m\times n$ matrix of output rank $r$. IterativeCUR relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds. We demonstrate through extensive experiments that IterativeCUR achieves up to $4\times$ speed-up over state-of-the-art pivoting-on-sketch approaches with no loss of accuracy, and up to $40\times$ speed-up over rank-adaptive randomized SVD approaches.

math.NA↗
arXiv2026

Multigrid with Linear Storage Complexity

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.

math.NA↗