1,022 research outputs found
Introduction to hyperbolic equations and fluid-structure interaction
In this semester project we deal with hyperbolic partial differential equations and Fluid-Structure Interactio
A posteriori error estimates for the Electric Field Integral Equation on polyhedra
We present a residual-based a posteriori error estimate for the Electric
Field Integral Equation (EFIE) on a bounded polyhedron. The EFIE is a
variational equation formulated in a negative order Sobolev space on the
surface of the polyhedron. We express the estimate in terms of
square-integrable and thus computable quantities and derive global lower and
upper bounds (up to oscillation terms).Comment: Submitted to Mathematics of Computatio
An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation
We consider a diffusion equation with highly oscillatory coefficients that
admits a homogenized limit. As an alternative to standard corrector problems,
we introduce here an embedded corrector problem, written as a diffusion
equation in the whole space in which the diffusion matrix is uniform outside
some ball of radius . Using that problem, we next introduce three
approximations of the homogenized coefficients. These approximations, which are
variants of the standard approximations obtained using truncated (supercell)
corrector problems, are shown to converge when . We also discuss
efficient numerical methods to solve the embedded corrector problem
A Scalable Two-Level Domain Decomposition Eigensolver for Periodic Schr\"odinger Eigenstates in Anisotropically Expanding Domains
Accelerating iterative eigenvalue algorithms is often achieved by employing a
spectral shifting strategy. Unfortunately, improved shifting typically leads to
a smaller eigenvalue for the resulting shifted operator, which in turn results
in a high condition number of the underlying solution matrix, posing a major
challenge for iterative linear solvers. This paper introduces a two-level
domain decomposition preconditioner that addresses this issue for the linear
Schr\"odinger eigenvalue problem, even in the presence of a vanishing
eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal
shift, which is already available as the solution to a spectral cell problem,
is required for the eigenvalue solver, it is logical to also use its associated
eigenfunction as a generator to construct a coarse space. We analyze the
resulting two-level additive Schwarz preconditioner and obtain a condition
number bound that is independent of the domain's anisotropy, despite the need
for only one basis function per subdomain for the coarse solver. Several
numerical examples are presented to illustrate its flexibility and efficiency.Comment: 30 pages, 7 figures, 2 table
- …