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

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

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

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

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

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

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

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