Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1509.02239

Mixed GMsFEM for the simulation of waves in highly heterogeneous media

Eric T. Chung·Wing Tat Leung
View original source ↗Read source document ↗Export metadata ↓

Abstract

Numerical simulations of waves in highly heterogeneous media have important applications, but direct computations are prohibitively expensive. In this paper, we develop a new generalized multiscale finite element method with the aim of simulating waves at a much lower cost. Our method is based on a mixed Galerkin type method with carefully designed basis functions that can capture various scales in the solution. The basis functions are constructed based on some local snapshot spaces and local spectral problems defined on them. The spectral problems give a natural ordering of the basis functions in the snapshot space and allow systematically enrichment of basis functions. In addition, by using a staggered coarse mesh, our method is energy conserving and has block diagonal mass matrix, which are desirable properties for wave propagation. We will prove that our method has spectral convergence, and present numerical results to show the performance of the method.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Eric T. Chung, Wing Tat Leung. 2015-09-08. Mixed GMsFEM for the simulation of waves in highly heterogeneous media. https://arxiv.org/abs/1509.02239

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

KEEP EXPLORING

Related papers

arXiv2026

The Stability of Block Eliminations and Additive Modifications

The block elimination with additive modifications (BEAM) method was recently proposed as a alternative to LU with partial pivoting requiring less communication. Because of the novelty of BEAM, the existing theoretical analysis is lacking. To that end, we analyze both the numerical stability of the underlying block LU factorization and the effects of additive modifications. For the block LU factorization, we are able to improve the previous results of Demmel et al. from being cubic in the element growth to merely quadratic. Furthermore, we propose an alternative measure of element growth that is better aligned with block LU; this new measure of growth allows our analysis to apply to matrices that cannot be factored with pointwise LU. In the second part, we analyzed the modifications produced by BEAM and the effect they have on the condition number and growth factor. Finally, we show that BEAM will not apply any modifications in some cases that regular block LU can safely factor.

math.NA↗
arXiv2026

Efficient Rigorous Continuation via Chebyshev Series Expansion I

We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted $\ell^1$ space, giving a finer control than the typical $C^0$-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn--Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada--Kawasaki--Teramoto system.

math.NA↗
arXiv2026

Efficient iterative techniques for solving tensor problems with the T-product

This paper develops two efficient iterative methods for solving tensor equations under the T-product framework. For T-symmetric positive definite tensor equations of the form $\mathcal{C} \star \mathcal{X} = \mathcal{D}$, we propose a conjugate-gradient-type algorithm that generates orthogonal residual and $\mathcal{C}$-orthogonal direction sequences, ensuring convergence within a finite number of steps. For general consistent tensor equations, we extend the method using a normal-equation transformation, and further adapt it to handle inconsistent systems by solving a least-squares minimization problem. Key advantages include direct tensor-based computations without explicit matrix expansion, rigorous finite-step convergence proofs, and the ability to obtain minimal Frobenius norm solutions. Numerical experiments on synthetic data, benchmark images, and video sequences demonstrate that the proposed algorithms achieve high precision with low computational time, confirming their practicality for large-scale multidimensional problems.

math.NA↗