Spectral Elements

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

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 N and the spectral degree p 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.

In this work a new approach to time dependent problems in combination with the Least-Squares Spectral Element Method (LSQSEM) will be discussed. Various timestepping formulations will be presented. These time-stepping formulations will be compared to the full space-time formulation. It will be shown that time-stepping formulations give accurate results for comparable CPU times. Furthermore is will be shown that a smaller timestep or a higher polynomial degree not always decreases the error norm.

In this work a new approach to time dependent problems in combination with the Least-Squares Spectral Element Method (LSQSEM) will be discussed. Various timestepping formulations will be presented. These time-stepping formulations will be compared to the full space-time formulation. It will be shown that time-stepping formulations give accurate results for comparable CPU times. Furthermore is will be shown that a smaller timestep or a higher polynomial degree not always decreases the error norm.

We present a new method for wave propagation in global earth models based upon the coupling between the spectral element method and a modal solution method. The Earth is decomposed into two parts, an outer shell with 3-D lateral heterogeneities and an inner sphere with only spherically symmetric heterogeneities. Depending on the problem, the outer heterogeneous shell can be mapped as the whole mantle or restricted only to the upper mantle or the crust. In the outer shell, the solution is sought in terms of the spectral element method, which stem from a high order variational formulation in space and a second-order explicit scheme in time. In the inner sphere, the solution is sought in terms of a modal solution in frequency after expansion on the spherical harmonics basis. The spectral element method combines the geometrical flexibility of finite element methods with the exponential convergence rate of spectral methods. It avoids the pole problems and allows for local mesh refinement, using a non-conforming discretization, for the resolution of sharp variations and topography along interfaces. The modal solution allows for an accurate isotropic representation in the inner sphere. The coupling is introduced within the spectral element method via a Dirichlet-to-Neumann (DtN) operator. The operator is explicitly constructed in frequency and in generalized spherical harmonics. The inverse transform in space and time requires special attention and an asymptotic regularization. The coupled method allows a significant speed-up in the simulation of the wave propagation in earth models. For spherically symmetric earth model, the method is shown to have the accuracy of spectral transform methods and allow the resolution of wavefield propagation, in 3-D laterally heterogeneous models, without any perturbation hypothesis.

Model generalized eigenproblems associated with self-adjoint differential operators in nonstandard homogeneous or heterogeneous domains are considered. Their numerical approximation is based on Gauss-Lobatto-Legendre conforming spectral elements defined by Gordon-Hall transfinite mappings. The resulting discrete eigenproblems are solved iteratively with a Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method, accelerated by an overlapping Schwarz preconditioner. Several numerical tests show the good convergence properties of the proposed preconditioned eigensolver, such as its scalability and quasi-optimality in the discretization parameters, which are analogous to those obtained for overlapping Schwarz preconditioners for linear systems.

A two-level overlapping domain decomposition method is analyzed for a Nédélec spectral element approximation of a model problem appearing in the solution of Maxwell's equations. The overlap between subdomains can consist of entire spectral elements or rectangular subsets of spectral elements. For fixed relative overlap and overlap made from entire elements, the condition number of the method is bounded, independently of the mesh size, the number of subregions, the coefficients and the degree of the spectral elements. In the case of overlap including just parts of spectral elements, a bound linear in the degree of the elements is proven. It is assumed that the coarse and fine mesh are quasi-uniform and shape-regular and that the domain is convex. Arguments that would not require quasi-uniformity of the coarse mesh and convexity of the domain are mentioned. Our work generalizes results obtained for lower-order Nédélec elements in Toselli [Numer. Math. (2000) 86:733-752]. Numerical results for the two-level algorithm in two dimensions are also presented, supporting our analysis.

We present an extension to the coupling scheme of the spectral element method (SEM) with a normal-mode solution in spherical geometry. This extension allows us to consider a thin spherical shell of spectral elements between two modal solutions above and below. The SEM is based on a high-order variational formulation in space and a second-order explicit scheme in time. It combines the geometrical flexibility of the classical finite-element method with the exponential convergence rate associated with spectral techniques. In the inner sphere and outer shell, the solution is sought in terms of a modal solution in the frequency domain after expansion on the spherical harmonics basis. The SEM has been shown to obtain excellent accuracy in solving the wave equation in complex media but is still numerically expensive for the whole Earth for high-frequency simulations. On the other hand, modal solutions are well known and numerically cheap in spherically symmetric models. By combining these two methods we take advantage of both, allowing high-frequency simulations in global Earth models with 3-D structure in a limited depth range. Within the spectral element method, the coupling is introduced via a dynamic interface operator, a Dirichlet-to-Neumann operator which can be explicitly constructed in the frequency and generalized spherical harmonics domain using modal solutions in the inner sphere and outer shell. The presence of the source and receivers in the top modal solution shell requires some special treatment. The accuracy of the method is checked against the mode summation method in simple spherically symmetric models and shows very good agreement for all type of waves, including diffracted waves travelling on the coupling boundary. A first simulation in a 3-D D-layer model based on the tomographic model SAW24b16 is presented up to a corner frequency of 1/12 s. The comparison with data shows surprisingly good results for the 3-D model even when the observed waveform amplitudes differ significantly from those predicted in the spherically symmetric reference model (PREM).

A spectral element (SE) implementation of the Givoli-Neta non-reflecting boundary condition (NRBC) is considered for the solution of the Klein-Gordon equation. The infinite domain is truncated via an artificial boundary B, and a high-order NRBC is applied on B. Numerical examples, in various configurations, concerning the propagation of a pressure pulse are used to demonstrate the performance of the SE implementation. Effects of time integration techniques and long term results are discussed. Specifically, we show that in order to achieve the full benefits of high-order accuracy requires balancing all errors involved; this includes the order of accuracy of the spatial discretization method, timeintegrators, and boundary conditions.

A reduced shallow water model under constant, non-zero advection in the infinite channel is considered. High-order (Givoli-Neta) non-reflecting boundary conditions are introduced in various configurations to create a finite computational space and solved using a spectral element formulation with high-order time integration. Numerical examples are used to demonstrate the synergy of using high-order spatial, time, and boundary discretization. We show that by balancing all numerical errors involved, high-order accuracy can be achieved for unbounded domain problems.

An efficient p-multigrid method is developed to solve the algebraic systems which result from the approximation of elliptic problems with the so-called Fekete-Gauss Spectral Element Method, which makes use of the Fekete points of the triangle as interpolation points and of the Gauss points as quadrature points. A multigrid strategy is defined by comparison of different prolongation/restriction operators and coarse grid algebraic systems. The efficiency and robustness of the approach, with respect to the type of boundary condition and to the structured/unstructured nature of the mesh, are highlighted through numerical examples.

Least-squares methods for partial differential equations are based on a normequivalence between the error norm and the residual norm. The resulting algebraic system of equations, which is symmetric positive definite, can also be obtained by solving a weighted collocation scheme using least-squares to solve the resulting algebraic equations. Furthermore, least-squares allows to ad extra constraints to the system. In the present work the entropy is added as an extra inequality constraint to ensure only physical solutions for the one-dimensional inviscid Burgers equation are obtained.

We present a hybrid spectral element/finite element domain decomposition method for solving elastic wave propagation problems. Aim of the method is to exploit both the enhanced accuracy of spectral elements, allowing significant reductions of the computational load, and the flexibility of finite elements in treating irregular geometries and nonlinear media. Interfaces between finite and spectral elements are managed by the mortar method, which enjoys optimal accuracy and allows to discretize the different regions independently. Several test cases illustrate the robustness of this method and its tolerance to a significant range of geometrical models, characterized by different grid refinements.

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 N and the spectral degree p 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.

We present a hybrid spectral element/finite element domain decomposition method for solving elastic wave propagation problems. Aim of the method is to exploit both the enhanced accuracy of spectral elements, allowing significant reductions of the computational load, and the flexibility of finite elements in treating irregular geometries and nonlinear media. Interfaces between finite and spectral elements are managed by the mortar method, which enjoys optimal accuracy and allows to discretize the different regions independently. Several test cases illustrate the robustness of this method and its tolerance to a significant range of geometrical models, characterized by different grid refinements.

In this paper we propose a mortar spectral element method for solving Maxwell's equations in 3D bounded cavities. The method is based on a non-conforming decomposition of the domain into the union of non-overlapping parallelepipeds. After proving an error estimate, we present some 3D computational results which confirm the performance of the method.

In the context of the numerical simulation of seismic wave propagation, the perfectly matched layer (PML) absorbing boundary condition has proven to be efficient to absorb surface waves as well as body waves with non grazing incidence. But unfortunately the classical discrete PML generates spurious modes traveling and growing along the absorbing layers in the case of waves impinging the boundary at grazing incidence. This is significant in the case of thin mesh slices, or in the case of sources located close to the absorbing boundaries or receivers located at large offset. In previous work we derived an unsplit convolutional PML (CPML) for staggered-grid finite-difference integration schemes to improve the efficiency of the PML at grazing incidence for seismic wave propagation. In this article we derive a variational formulation of this CPML method for the seismic wave equation and validate it using the spectral-element method based on a hybrid first/secondorder time integration sch...

We present in this paper a stable spectral element for the approximations of the grad(div) eigenvalue problem in two and three-dimensional quadrangular geometry. Spectral approximations based on Gaussian quadrature rules are built in a dual variational approach with Darcy type equations. We prove that spectral convergence can be reached for the irrotational spectrum without the presence of any spurious eigenmodes, provided an adequate choice is made for the quadrature rules.

The spectral element method can be used to deal with the spatial operators of neutron transport problems with high efficiency, as shown recently in the framework of the second-order A N transport approximation. The results highlight interesting computational features and show the appeal of the scheme for reactor physics applications. In this paper we investigate the numerical performance of the method in detail. In order to carry out an accurate monitoring of the error behavior to levels close to numerical round-off, we use benchmark problems with known analytical solutions, or with manufactured solutions. Manufactured solutions can easily be obtained for source-injected problems, by tailoring the external neutron source and the boundary conditions to a pre-established analytical solution for a given system. The results presented prove the effectiveness of the method and the high level of accuracy that can be attained.

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization.

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.

A numerical approximation of the acoustic wave equation is presented. The spatial discretization is based on conforming spectral elements, whereas we use finite difference Newmark's explicit integration schemes for the temporal discretization. A rigorous stability analysis is developed for the discretized problem providing an upper bound for the time step t. We present several numerical results concerning stability and convergence properties of the proposed numerical methods.

Model generalized eigenproblems associated with self-adjoint differential operators in nonstandard homogeneous or heterogeneous domains are considered. Their numerical approximation is based on Gauss-Lobatto-Legendre conforming spectral elements defined by Gordon-Hall transfinite mappings. The resulting discrete eigenproblems are solved iteratively with a Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method, accelerated by an overlapping Schwarz preconditioner. Several numerical tests show the good convergence properties of the proposed preconditioned eigensolver, such as its scalability and quasi-optimality in the discretization parameters, which are analogous to those obtained for overlapping Schwarz preconditioners for linear systems.

Abstract. In this work a new approach to time dependent problems in combination with the Least-Squares Spectral Element Method (LSQSEM) will be discussed. Various time-stepping formulations will be presented. These time-stepping formulations will be compared to the full space-time formulation. It will be shown that time-stepping formulations give accurate results for comparable CPU times. Furthermore is will be shown that a smaller timestep or a higher polynomial degree not always decreases the error norm. 1

Abstract. This papers describes the use of the Least-Squares Spectral Element Method to polynomial Chaos to solve stochastic partial dierential equations. The method will be described in detail and a comparison will be presented between the least-squares projection and the conventional Galerkin projection. 1