860 research outputs found
From Raviart-Thomas to HDG
This document has been motivated by a course entitled {\em From
Raviart-Thomas to HDG}, prepared for {\em C\'adiz Num\'erica 2013 -- Course and
Encounter on Numerical Analysis} (C\'adiz, Spain -- June 2013). It is an
introduction to the techniques for local analysis of classical mixed methods
for diffusion problems and how they motivate the Hybridizable Discontinuous
Galerkin method.Comment: 50 page
Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations
In this paper we present a battery of results related to how Galerkin
semidiscretization in space affects some formulations of wave scattering and
propagation problems when retarded boundary integral equations are used
Variational views of stokeslets and stresslets
In this paper we present a self-contained variational theory of the layer
potentials for the Stokes problem on Lipschitz boundaries. We use these weak
definitions to show how to prove the main theorems about the associated
Calder\'on projector. Finally, we relate these variational definitions to the
integral forms. Instead of working these relations from scratch, we show some
formulas parametrizing the Stokes layer potentials in terms of those for the
Lam\'e and Laplace operators. While all the results in this paper are well
known for smooth domains, and most might be known for non-smooth domains, the
approach is novel a gives a solid structure to the theory of Stokes layer
potentials
A fully discrete BEM-FEM scheme for transient acoustic waves
We study a symmetric BEM-FEM coupling scheme for the scattering of transient
acoustic waves by bounded inhomogeneous anisotropic obstacles in a homogeneous
field. An incident wave in free space interacts with the obstacles and produces
a combination of transmission and scattering. The transmitted part of the wave
is discretized in space by finite elements while the scattered wave is reduced
to two fields defined on the boundary of the obstacles and is discretized in
space with boundary elements. We choose a coupling formulation that leads to a
symmetric system of integro-differential equations. The retarded boundary
integral equations are discretized in time by Convolution Quadrature, and the
interior field is discretized in time with the trapezoidal rule. We show that
the scattering problem generates a C_0 group of isometries in a Hilbert space,
and use associated estimates to derive stability and convergence results. We
provide numerical experiments and simulations to validate our results and
demonstrate the flexibility of the method
New analytical tools for HDG in elasticity, with applications to elastodynamics
We present some new analytical tools for the error analysis of hybridizable
discontinuous Galerkin (HDG) method for linear elasticity. These tools allow us
to analyze more variants of HDG method using the projection-based approach,
which renders the error analysis simple and concise. The key result is a
tailored projection for the Lehrenfeld-Sch\"{o}berl type HDG (HDG+ for
simplicity) methods. By using the projection we recover the error estimates of
HDG+ for steady-state and time-harmonic elasticity in a simpler analysis. We
also present a semi-discrete (in space) HDG+ method for transient elastic waves
and prove it is uniformly-in-time optimal convergent by using the
projection-based error analysis. Numerical experiments supporting our analysis
are presented at the end.Comment: Submitted to Math. Comp. on 3/27/2019, accepted on 10/20/201
Convolution Quadrature for Wave Simulations
These notes build an introduction to Convolution Quadrature techniques
applied to linear convolutions and convolution equations with a bias to
problems related to wave propagation. The notes are self-contained and
emphasize algorithmic aspects. They include introductory material on
vector-valued distributions, convolution operators, and Dunford calculus
The Costabel-Stephan system of Boundary Integral Equations in the Time Domain
In this paper we formulate a transmission problem for the transient acoustic
wave equation as a system of retarded boundary integral equations. We then
analyse a fully discrete method using a general Galerkin
semidiscretization-in-space and Convolution Quadrature in time. All proofs are
developed using recent techniques based on the theory of evolution equations.
Some numerical experiments are provided
Brushing up a theorem by Lehel Banjai on the convergence of Trapezoidal Rule Convolution Quadrature
This document is made up of two different units. One of them is a regular
terse research article, whereas the other one is the detailed and independently
written explanations for the paper, so that readers of the short paper do not
need to go over all the cumbersome computations. The goal is to clarify the
dependence with respect to the time variable of some estimates about the
convergence of the Trapezoidal Rule based Convolution Quadrature method applied
to hyperbolic problems. This requires a careful investigation of the article of
Lehel Banjai where the first convergence estimates were introduced, and of some
technical results from a classical paper of Christian Lubich.Comment: 32 pages, 1 figure; First part of the article will be submitted to
Computers & Mathematics with Application
Some properties of layer potentials and boundary integral operators for the wave equation
In this work we establish some new estimates for layer potentials of the
acoustic wave equation in the time domain, and for their associated retarded
integral operators. These estimates are proven using time-domain estimates
based on theory of evolution equations and improve known estimates that use the
Laplace transform.Comment: 29 page
Boundary-Finite Element discretization of time dependent acoustic scattering by elastic obstacles with piezoelectric behavior
A coupled BEM/FEM formulation for the transient interaction between an
acoustic field and a piezoelectric scatterer is proposed. The scattered part of
the acoustic wave is represented in terms of retarded layer potentials while
the elastic displacement and electric potential are treated variationally. This
results in an integro-differential system. Well posedness of a general Galerkin
semi-discretization in space of the problem is shown in the Laplace domain and
translated into explicit stability bounds in the time domain. Trapezoidal-Rule
and BDF2 Convolution Quadrature are used in combination with matching time
stepping for time discretization. Second order convergence is proven for the
BDF2-based method. Numerical experiments are provided for BDF2 and Trapezoidal
Rule based time evolution
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