Search

The problem of determining the manner in which an incoming acoustic wave is scattered by an elastic body immersed in a fluid is one of the central importance in detecting and identifying submerged objects. The problem is generally referred to as a fluid-structure interaction and is mathematically formulated as a time-dependent transmission problem. In this paper, we consider a typical fluid-structure interaction problem by using a coupling procedure that reduces the problem to a nonlocal initial-boundary problem in the elastic body with a system of integral equations on the interface between the domains occupied by the elastic body and the fluid. We analyze this nonlocal problem by the Lubich approach via the Laplace transform, an essential feature of which is that it works directly on data in the time domain rather than in the transformed domain. Our results may serve as a mathematical foundation for treating time-dependent fluid-structure interaction problems by convolution quadrature coupling of FEM and BEM. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We show some improved mapping properties of the Time Domain Electric Field Integral Equation and of its Galerkin semidiscretization in space. We relate the weak distributional framework with a stronger class of solutions using a group of strongly continuous operators. The stability and error estimates we derive are sharper than those in the literature. [ABSTRACT FROM AUTHOR]

We propose time-domain boundary integral and coupled boundary integral and variational formulations for acoustic scattering by linearly elastic obstacles. Well-posedness along with stability and error bounds with explicit time dependence are established. Full discretization is achieved coupling boundary and finite elements; convolution quadrature (CQ) is used for time evolution in the pure boundary integral equation formulation and combined with time stepping in the coupled boundary-finite element method (BEM-FEM) scenario. Second-order convergence in time is proved for BDF2-CQ and numerical experiments are provided for both BDF2 and trapezoidal rule CQ showing second-order behaviour for the latter as well. [ABSTRACT FROM AUTHOR]

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 (CQ) in time. All proofs are developed using recent techniques based on the theory of evolution equations. Some numerical experiments are provided. [ABSTRACT FROM AUTHOR]

In this article, we provide some MATLAB tools for efficient vectorized implementation of the Hybridizable Discontinuous Galerkin for linear variable coefficient reaction-diffusion problems in polyhedral domains. The resulting tools are modular and include enhanced structures to deal with convection-diffusion problems, plus several projection operators and the postprocessing implementation that is necessary to realize the superconvergence property of the method. Loops over the elements are exclusively local and, as such, have been parallelized. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a systematic way to obtain convergent finite element schemes for the Stokes-Darcy flow problem by combining well-known mixed finite elements that are separately convergent for Stokes and Darcy problems. In the approach in which the Darcy problem is set in its natural H(div) formulation and the Stokes problem is expressed in velocity0pressure form, the transmission condition ensuring global mass conservation becomes essential. As opposed to the strategy that handles weakly this transmission condition through a Lagrange multiplier, we impose here this restriction exactly in the global velocity space. Our analysis of the Galerkin discretization of the resulting problem reveals that if the mixed finite element space used in the Darcy domain admits an H(div)-stable discrete lifting of the normal trace, then it can be combined with any stable Stokes mixed finite element of the same order to deliver a stable global method with quasi-optimal convergence rate. Finally, we present a series of numerical tests confirming our theoretical convergence estimates. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyse a fully discrete method for a direct boundary integral formulation of the scattering of a transient acoustic wave by a sound-soft obstacle. The method uses the Galerkin Boundary Element Method in the space variables and three different choices of time-stepping strategies based on convolution quadrature. The numerical analysis of the method is carried out directly in the time domain, not reverting to Laplace transform techniques. [ABSTRACT FROM AUTHOR]

In this paper, we show that by sending the normal stabilization function to infinity in the hybridizable discontinuous Galerkin methods previously proposed in [Comput. Methods Appl. Mech. Engrg. 199 (2010), 582–597], for Stokes flows, a new class of divergence-conforming methods is obtained which maintains the convergence properties of the original methods. Thus, all the components of the approximate solution, which use polynomial spaces of degree k, converge with the optimal order of k + 1 in L2 for any k ≥ 0. Moreover, the postprocessed velocity approximation is also divergenceconforming, exactly divergence-free and converges with order k + 2 for k ≥ 1 and with order 1 for k = 0. The novelty of the analysis is that it proceeds by taking the limit when the normal stabilization goes to infinity in the error estimates recently obtained in [Math. Comp., 80 (2011) 723–760]. [ABSTRACT FROM AUTHOR]

This paper is concerned with a dual-mixed formulation for a three dimensional exterior Stokes problem via boundary integral equation methods. Here velocity, pressure and stress are the main unknowns. Following a similar analysis given recently for the Laplacian, we are able to extend the classical Johnson & Nédélec procedure to the present case, without assuming any restrictive smoothness requirement on the coupling boundary, but only Lipschitz-continuity. More precisely, after using the incompressibility condition to eliminate the pressure, we consider the resulting velocity-stress approach with a Neumann boundary condition on an annular bounded domain, and couple the underlying equations with only one boundary integral equation arising from the application of the normal trace to the Green representation formula in the exterior unbounded region. As a result, we obtain a saddle point operator equation, which is then analyzed by the wellknown Babuška-Brezzi theory: in particular, the well-posedness of the formulation will be established. [ABSTRACT FROM AUTHOR]

In this short article we prove that the classical one-equation (or Johnson-Nédélec's) coupling of finite and boundary elements can be applied with a Lipschitz coupling interface. Because of the way it was originally approached from the analytical standpoint, this BEM-FEM scheme has generally required smooth boundaries and hence produced a consistency error in the finite element part. With a variational argument, we prove that this requirement is not needed and that stability holds for all pairs of discrete space, as it inherits the underlying ellipticity of the problem. [ABSTRACT FROM AUTHOR]

We present new a priori error analyses of the coupling of the Raviart--Thomas (RT) method and BEM as well as of the coupling of the hybridizable discontinuous Galerkin (HDG) method with BEM. The novel features of the analysis of the RT-BEM coupling are the superconvergence estimates for the scalar approximation of the interior problem and new bounds in weak and strong norms for the boundary variables. The analysis of the HDG-BEM coupling is the first analysis of this coupling and shows that the coupling provides approximations with the same convergence properties as those of the RT-BEM coupling. [ABSTRACT FROM AUTHOR]

We propose a general framework that allows for a new natural coupling of boundary element and a wide class of finite element methods (FEMs) for a model second-order elliptic problem. This class of FEMs includes mixed methods, discontinuous Galerkin methods and the continuous Galerkin method. We provide sufficient conditions guaranteeing the well-posedness of the methods and give several examples that include new as well as old methods. [ABSTRACT FROM PUBLISHER]

In this paper we develop the a priori and a posteriori error analyses of a mixed finite element method for the coupling of fluid flow with nonlinear porous media flow. Flows are governed by the Stokes and nonlinear Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces and the Beavers–Joseph–Saffman law. We consider dual-mixed formulations in both domains, and in order to handle the nonlinearity involved, we introduce the pressure gradient in the Darcy region as an auxiliary unknown. In addition, the transmission conditions become essential, which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. As a consequence, the resulting variational formulation can be written, conveniently, as a twofold saddle point operator equation. Thus, a well-known generalization of the classical Babuška–Brezzi theory is applied to show the well-posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimate. In particular, the set of feasible finite element subspaces includes Raviart–Thomas elements of lowest order and piecewise constants for the velocities and pressure, respectively, in both domains, together with piecewise constant vectors for the Darcy pressure gradient and continuous piecewise linear elements for the traces. Then, we employ classical approaches and use known estimates to derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Finally, several numerical results confirming the good performance of the method and the theoretical properties of the a posteriori error estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution, are reported. [ABSTRACT FROM PUBLISHER]

In this paper, we apply the coupling of local discontinuous Galerkin and boundary element methods to solve a class of non-linear exterior transmission problems in the plane. As a model, we consider a non-linear elliptic equation in an annular polygonal domain coupled with the Poisson equation in the surrounding unbounded exterior region. In addition, we assume discontinuous transmission conditions on the interface boundary. Our approach constitutes an extension to non-linear problems of the a priori error analysis developed recently for linear exterior transmission problems. Most of the techniques employed here are similar to the linear case but some differences appear. In particular, because of the non-homogeneous transmission conditions, the so-called numerical fluxes need to be suitably defined. We prove stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Then, we apply local and global approximation properties of the discrete spaces to obtain the a priori error estimates in the energy norm. [ABSTRACT FROM PUBLISHER]

In this article we generalize some standard local approximation properties of piecewise polynomials, which are measured only in the Sobolev seminorms of order 0 and 1, to the case of fractional Sobolev seminorms. The simplicity of the corresponding proofs is an interesting feature of the presentation. In addition, our analysis makes clear at each step that the constants involved are independent of the meshsizes. As an application, we show that the corresponding generalized estimates simplify the a priori error analysis of the local discontinuous Galerkin methods in the case of general non-convex polyhedral domains. [ABSTRACT FROM AUTHOR]

We derive and analyze a hybridizable discontinuous Galerkin (HDG) method for approximating weak solutions to the equations of time-harmonic linear elasticity on a bounded Lipschitz domain in three dimensions. The real symmetry of the stress tensor is strongly enforced and its coefficients as well as those of the displacement vector field are approximated simultaneously at optimal convergence with respect to the choice of approximating spaces, wavenumber, and mesh size. Sufficient conditions are given so that the system is indeed transferable onto a global hybrid variable that, for larger polynomial degrees, may be approximated via a smaller-dimensional space than the original variables. We construct several variants of this method and discuss their advantages and disadvantages, and give a systematic approach to the error analysis for these methods. We touch briefly on the application of this error analysis to the time-dependent problem, and finally, we examine two different implementations of the method over various polynomial degrees and numerically demonstrate the convergence properties proven herein. [ABSTRACT FROM AUTHOR]

We propose and analyze a new coupling procedure for the Hybridizable Discontinuous Galerkin Method with Galerkin Boundary Element Methods based on a double layer potential representation of the exterior component of the solution of a transmission problem. We show a discrete uniform coercivity estimate for the non-symmetric bilinear form and prove optimal convergence estimates for all the variables, as well as superconvergence for some of the discrete fields. Some numerical experiments support the theoretical findings. [ABSTRACT FROM AUTHOR]

The existence of uniformly bounded discrete extension operators is established for conforming Raviart-Thomas and Nédelec discretizations of H(div) and H(curl) on locally refined partitions of a polyhedral domain into tetrahedra. [ABSTRACT FROM AUTHOR]

In this paper, we present a fully discretized Calderón Calculus for the two dimensional Helmholtz equation. This full discretization can be understood as highly non-conforming Petrov-Galerkin methods, based on two staggered grids of mesh size h, Dirac delta distributions substituting acoustic charge densities and piecewise constant functions for approximating acoustic dipole densities. The resulting numerical schemes from this calculus are all of order h2 provided that the continuous equations are well posed. We finish by presenting some numerical experiments illustrating the performance of this discrete calculus. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present and test a full discretization of all elements of the Calderón Calculus (layer potentials and integral operators) for the Helmholtz equation in smooth closed curves in the plane. The resulting integral equations provide approximations of order three for all variables involved. Tests are shown for a wide array of direct, indirect and combined field integral equation at fixed frequency and for a Convolution Quadrature based approximation in the time domain. [Copyright &y& Elsevier]

We introduce a least squares method for discretizing the time-harmonic Maxwell equations written as a first order system. The method belongs to the recently introduced least squares approach called the Saddle Point Least Squares (SPLS) method (see Bacuta and Qirko 2015, 2017). It is related to two known methods: the Bramble and Pasciak least squares method for solving electromagnetic problems (Bramble and Pasciak, 2004; Bramble et al., 2005) and the minimum residual DPG method of Demkowicz and Gopalakrishnan (2010, 2013). The proposed discretization is based on reduction to a symmetric saddle point formulation on spaces of piecewise polynomial functions. The edge elements are avoided. A discrete test space is chosen first among standard conforming finite element spaces and the discrete trial space is chosen second as the image of the first order Maxwell differential operator on the discrete test space. For the proposed iterative processes, a nodal basis for the trial space is not required and the main inversion at each step is done only on the test space. The method is efficient for both convex and non-convex domains and for a various range of frequencies as shown by the numerical results. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a full discretization of the exterior transient Stokes problem with Dirichlet boundary conditions. The method is based on a single layer boundary integral representation, using Galerkin semidiscretization in the space variables and multistep convolution quadrature in time. Convergence estimates are based on a Laplace domain analysis, which translates into a detailed study of the exterior Brinkman problem. Some numerical experiments are provided [ABSTRACT FROM AUTHOR]

We consider the approximation of an abstract evolution problem with inhomogeneous side constraint using A -stable Runge-Kutta methods. We derive a priori estimates in norms other than the underlying Banach space. Most notably, we derive estimates in the graph norm of the generator. These results are used to study convolution quadrature based discretizations of a wave scattering and a heat conduction problem., (© The Author(s) 2020.)