We present a parallel preconditioning method for the iterative solution of thetime-harmonic elastic wave equation which makes use of higher-order spectral elements toreduce pollution error. In particular, the method leverages perfectly matched layerboundary conditions to efficiently approximate the Schur complement matrices of a blockLDLTfactorization. Both sequential and parallel versions of the algorithm are discussed andresults for large-scale problems from exploration geophysics are presented.

We consider the flow of a viscous incompressible fluid in a rigid homogeneous porousmedium provided with mixed boundary conditions. Since the boundary pressure can presenthigh variations, the permeability of the medium also depends on the pressure, so that themodel is nonlinear. A posteriori estimates allow us to omit thisdependence where the pressure does not vary too much. We perform the numerical analysis ofa spectral element discretization of the simplified model. Finally we propose a strategywhich leads to an automatic identification of the part of the domain where the simplifiedmodel can be used without increasing significantly the error.

Spectral element methods on simplicial meshes, say TSEM, show both the advantages of spectral and finite element methods, i.e., spectral accuracy and geometrical flexibility. We present aTSEM solver of the two-dimensional (2D) incompressible Navier-Stokes equations, with possible extension to the 3D case. It uses a projection method in time and piecewise polynomial basis functions of arbitrary degree in space. The so-called Fekete-Gauss TSEM is employed,i.e., Fekete (resp. Gauss) points of the triangle are used as interpolation (resp. quadrature) points. For the sake of consistency, isoparametric elements are used to approximate curved geometries. The resolution algorithm is based on an efficient Schur complement method, so that one only solves for the element boundary nodes. Moreover, the algebraic system is never assembled, therefore the number of degrees of freedom is not limiting. An accuracy study is carried out and results are provided for classical benchmarks: the driven cavity flow, the flow between eccentric cylinders and the flow past a cylinder.

The additive Schwarz preconditioner with minimal overlap is extended to triangular spectral elements (TSEM). The method is a generalization of the corresponding method in tensorial quadrilateral spectral elements (QSEM). The proposed preconditioners are based on partitioning the domain into overlapping subdomains, solving local problems on these subdomains and solving an additional coarse problem associated with the subdomain mesh. The results of numerical experiments show that the proposed preconditioner are robust with respect to the number of elements and are more efficient than the preconditioners with generous overlaps.

In this paper we study Bénard-Marangoni convection in confined containers where a thin fluid layer is heated from below. We consider containers with circular, square and hexagonal cross-sections. For Marangoni numbers close to the critical Marangoni number, the flow patterns are dominated by the appearance of the well-known hexagonal convection cells. The main purpose of this computational study is to explore the possible patterns the system may end up in for a given set of parameters. In a series of numerical experiments, the coupled fluid-thermal system is started with a zero initial condition for the velocity and a random initial condition for the temperature. For a given set of parameters we demonstrate that the system can end up in more than one state. For example, the final state of the system may be dominated by a steady convection pattern with a fixed number of cells, however, the same system may occasionally end up in a steady pattern involving a slightly different number of cells, or it may end up in a state where most of the cells are stationary, while one or more cells end up in an oscillatory state. For larger aspect ratio containers, we are also able to reproduce dislocations in the convection pattern, which have also been observed experimentally. It has been conjectured that such imperfections (e.g., a localized star-like pattern) are due to small irregularities in the experimental setup (e.g., the geometry of the container). However, we show, through controlled numerical experiments, that such phenomena may appear under otherwise ideal conditions. By repeating the numerical experiments for the same non-dimensional numbers, using a different random initial condition for the temperature in each case, we are able to get an indication of how rare such events are. Next, we study the effect of symmetrizing the initial conditions. Finally, we study the effect of selected geometry deformations on the resulting convection patterns.

A number of approaches for discretizing partial differential equations with random dataare based on generalized polynomial chaos expansions of random variables. These constitutegeneralizations of the polynomial chaos expansions introduced by Norbert Wiener toexpansions in polynomials orthogonal with respect to non-Gaussian probability measures. Wepresent conditions on such measures which imply mean-square convergence of generalizedpolynomial chaos expansions to the correct limit and complement these with illustrativeexamples.

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

This paper deals with the mortar spectral element discretization of two equivalent problems, the Laplace equation and the Darcy system, in a domain which corresponds to a nonhomogeneous anisotropic medium. The numerical analysis of the discretization leads to optimal error estimates and the numerical experiments that we present enable us to verify its efficiency.

In this paper, we present extensive numerical tests showing the performanceand robustness of a Balancing Neumann-Neumann method for the solution of algebraic linear systems arising from hp finite element approximations of scalar ellipticproblems on geometrically refined boundary layer meshes inthree dimensions. The numerical results are in good agreement with the theoretical bound for the condition number of the preconditioned operator derived in [Toselli and Vasseur, IMA J. Numer. Anal.24 (2004) 123–156]. They confirm that the condition numbers are independent of theaspect ratio of the mesh and of potentially large jumps of thecoefficients. Good results are also obtained for certain singularly perturbed problems. The condition numbers only grow polylogarithmically with thepolynomial degree, as in the case of p approximations on shape-regular meshes [Pavarino, RAIRO: Modél. Math. Anal. Numér.31 (1997) 471–493]. This paper follows [Toselli and Vasseur, Comput. Methods Appl. Mech. Engrg.192 (2003) 4551–4579] on two dimensional problems.