Elena Zampieri

Generalized Galerkin approximations of elastic waves with absorbing boundary conditions

Abstract For the propagation of elastic waves in unbounded domains, absorbing boundary conditions (ABCs) at the fictitious numerical boundaries have been proposed. In this paper we focus on both first- and second-order ABCs in the framework of variational (weak) approximations, like those stemming from Galerkin method (or its variants) for finite element or spectral approximations [1]. In particular, we recover first order conditions as natural (or Neumann) conditions, whereas we propose a penalty residual method for the treatment of second order ABCs. The time discretization is based on implicit backward finite differences, whereas we use spectral Legendre collocation methods set in a variational form for the spatial discretization (treatment of finite element or spectral element approximations is completely similar). Numerical experiments exhibit that the present formulation of second-order ABCs improves the one based on first-order ABCs with regard to both the reduction of the total energy in the computational domain, and the Fourier spectrum of the displacement field at selected points of the elastic medium. A stability analysis is developed for the variational problem in the continuous case both for first- and second-order ABCs. A suitable treatment of ABCs at corners is also proposed.

Finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems

Spectral collocation approximations based on Legendre–Gauss–Lobatto (LGL) points for Helmholtz equations as well as for the linear elasticity system in rectangular domains are studied.The collocati...

Overlapping Schwarz Methods for Fekete and Gauss-Lobatto Spectral Elements

The classical overlapping Schwarz algorithm is here extended to the triangular/tetrahedral spectral element (TSEM) discretization of elliptic problems. This discretization, based on Fekete nodes, is a generalization to nontensorial elements of the tensorial Gauss-Lobatto-Legendre quadrilateral spectral elements (QSEM). The overlapping Schwarz preconditioners are based on partitioning the domain of the problem into overlapping subdomains, solving local problems on these subdomains, and solving an additional coarse problem associated with either the subdomain mesh or the spectral element mesh. The overlap size is generous, i.e., one element wide, in the TSEM case, while it is minimal or variable in the QSEM case. The results of several numerical experiments show that the convergence rate of the proposed preconditioning algorithm is independent of the number of subdomains

An explicit second order spectral element method for acoustic waves

N

Numerical approximation of elastic waves equations by implicit spectral methods

and the spectral degree

Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves

p

Numerical solution of linear elastic problems by spectral collocation methods

in case of generous overlap; otherwise it depends inversely on the overlap size. The proposed preconditioners are also robust with respect to arbitrary jumps of the coefficients of the elliptic operator across subdomains.

Overlapping Schwarz Preconditioners for Fekete Spectral Elements

Abstract The acoustic wave equation is here discretized by conforming spectral elements in space and by the second order leap-frog method in time. For simplicity, homogeneous boundary conditions are considered. A stability analysis of the resulting method is presented, providing an upper bound for the allowed time step that is proportional to the size of the elements and inversely proportional to the square of their polynomial degree. A convergence analysis is also presented, showing that the convergence error decreases when the time step or the size of the elements decrease or when the polynomial degree increases. Several numerical results illustrating these results are presented.

On the condition number of some spectral collocation operators and their finite element preconditioning

A numerical treatment of the linear elasto-dynamic problem is presented. The spatial discretization is based on Legendre collocation methods (also called pseudo-spectral methods) set in weak form, whereas we use implicit integration schemes based on finite differences for the temporal discretization. Absorbing boundary conditions are introduced in order to face wave propagation problems in unbounded domains. We present several numerical results concerning stability, convergence, absorption properties for waves leaving the boundaries, and comparison between explicit and implicit time advancing schemes. The main result of the paper is as follows: the implicit advancing time scheme allows us to adopt an integration step time about ten times larger than the one obtained by assuming an explicit advancing time scheme.

Preconditioners for spectral discretizations of Helmholtz's equation with Sommerfeld boundary conditions ☆

The classical overlapping Schwarz algorithm is here extended to the spectral element discretization of linear elastic problems, for both homogeneous and heterogeneous compressible materials. The algorithm solves iteratively the resulting preconditioned system of linear equations by the conjugate gradient or GMRES methods. The overlapping Schwarz preconditioned technique is then applied to the numerical approximation of elastic waves with spectral elements methods in space and implicit Newmark time advancing schemes. The results of several numerical experiments, for both elastostatic and elastodynamic problems, show that the convergence rate of the proposed preconditioning algorithm is independent of the number of spectral elements (scalability), is independent of the spectral degree in case of generous overlap, otherwise it depends inversely on the overlap size. Some results on the convergence properties of the spectral element approximation combined with Newmark schemes for elastic waves are also presented.