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We present and study a novel numerical algorithm to approximate the action of Tβ := L-β where L is a symmetric and positive definite unbounded operator on a Hilbert space H0. The numerical method is based on a representation formula for T-β in terms of Bochner integrals involving (I + t2L)-1 for t ∊ (0,∞). To develop an approximation to Tβ, we introduce a finite element approximation Lh to L and base our approximation to Tβ on Tβ h := L -β h . The direct evaluation of Tβ h is extremely expensive as it involves expansion in the basis of eigenfunctions for Lh. The above mentioned representation formula holds for T -β h and we propose three quadrature approximations denoted generically by Qβ h. The two results of this paper bound the errors in the H0 inner product of Tβ -Tβ h πh and Tβ h -Qβ h where πh is the H0 orthogonal projection into the finite element space. We note that the evaluation of Qβ h involves application of (I +(ti)2Lh)-1 with ti being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error Tβ h - Qβ h and the finite element error Tβ - Tβ h πh. [ABSTRACT FROM AUTHOR]

We study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off-diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: We consider the application of a perfectly matched layer (PML) technique applied in Cartesian geometry to approximate solutions of the acoustic scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift (“stretching”) and leads to a variable complex coefficient equation for the acoustic wave posed on an infinite domain, the complement of the bounded scatterer. The use of Cartesian geometry leads to a PML operator with simple coefficients, although, still complex symmetric (non-Hermitian). The PML reformulation results in a problem whose solution coincides with the original solution inside the PML layer while decaying exponentially outside. The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem). This paper provides new stability estimates for the Cartesian PML approximations both on the infinite and the truncated domain. We first investigate the stability of the infinite PML approximation as a function of the PML strength . This is done for PML methods which involve continuous piecewise smooth stretching as well as piecewise constant stretching functions. We next introduce a truncation parameter which determines the size of the PML layer. Our analysis shows that the truncated PML problem is stable provided that the product of is sufficiently large, in which case the solution of the problem on the truncated domain converges exponentially to that of the original problem in the domain of interest near the scatterer. This justifies the simple computational strategy of selecting a fixed PML layer and increasing to obtain the desired accuracy. The results of numerical experiments varying and are given which illustrate the theoretically predicted behavior. [Copyright &y& Elsevier]

We consider the application of a perfectly matched layer (PML) technique applied in Cartesian geometry to approximate solutions of the electromagnetic wave (Maxwell) scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift ("stretching") and leads to a variable complex coefficient equation for the electric field posed on an infinite domain, the complement of a bounded scatterer. The use of Cartesian geometry leads to a PML operator with simple coefficients, although, still complex symmetric (non-Hermitian). The PML reformulation results in a problem which preserves the original solution inside the PML layer while decaying exponentially outside. The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem). For suitably defined Cartesian PML layers, we prove existence and uniqueness of the solutions to the infinite domain and truncated domain PML equations provided that the truncated domain is sufficiently large. We show that the PML reformulation preserves the solution in the layer while decaying exponentially outside of the layer. Our approach is to develop variational stability for the infinite domain electromagnetic wave scattering PML problem from that for the acoustic wave (Helmholtz) scattering PML problem given in [12]. The stability and exponential convergence of the truncated PML problem is then proved using the decay properties of solutions of the infinite domain problem. Although, we do not provide a complete analysis of the resulting finite element approximation, we believe that such an analysis should be possible using the techniques in [6]. [ABSTRACT FROM AUTHOR]

We design and analyze variational and non-variational multigrid algorithms for the Laplace-Beltrami operator on a smooth and closed surface. In both cases, a uniform convergence for the V -cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid. The main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface. In addition, the vanishing mean value constraint is imposed on each level, thereby avoiding singular quadratic forms without adding additional computational cost. Numerical results supporting our analysis are reported. In particular, the algorithms perform well even when applied to surfaces with a large aspect ratio. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a Cartesian PML approximation to solutions of acoustic scattering problems on an unbounded domain in . The perfectly matched layer (PML) technique in a curvilinear coordinate system has been researched for acoustic scattering applications both in theory and computation. Our goal will be to extend the results of spherical/cylindrical PML to PML in Cartesian coordinates, that is, the well-posedness of Cartesian PML approximation on both the unbounded and truncated domains. The exponential convergence of approximate solutions as a function of domain size is also shown. We note that once the stability and convergence of the (continuous) truncated problem has been achieved, the analysis of the resulting finite element approximations is then classical. Finally, the results of numerical computations illustrating the theory and efficiency of the Cartesian PML approach will be given. [Copyright &y& Elsevier]

Abstract: In this paper, we study the spectrum of the operator which results when the Perfectly Matched Layer (PML) is applied in Cartesian geometry to the Laplacian on an unbounded domain. This is often thought of as a complex change of variables or “complex stretching.” The reason that such an operator is of interest is that it can be used to provide a very effective domain truncation approach for approximating acoustic scattering problems posed on unbounded domains. Stretching associated with polar or spherical geometry lead to constant coefficient operators outside of a bounded transition layer and so even though they are on unbounded domains, they (and their numerical approximations) can be analyzed by more standard compact perturbation arguments. In contrast, operators associated with Cartesian stretching are non-constant in unbounded regions and hence cannot be analyzed via a compact perturbation approach. Alternatively, to show that the scattering problem PML operator associated with Cartesian geometry is stable for real nonzero wave numbers, we show that the essential spectrum of the higher order part only intersects the real axis at the origin. This enables us to conclude stability of the PML scattering problem from a uniqueness result given in a subsequent publication. [Copyright &y& Elsevier]

Abstract: In this note, we provide existence and uniqueness results for frequency domain elastic wave problems. These problems are posed on the complement of a bounded domain (the scatterer). The boundary condition at infinity is given by the Kupradze–Sommerfeld radiation condition and involves different Sommerfeld conditions on different components of the field. Our results are obtained by setting up the problem as a variational problem in the Sobolev space on a bounded domain. We use a nonlocal boundary condition which is related to the Dirichlet to Neumann conditions used for acoustic and electromagnetic scattering problems. We obtain stability results for the source problem, a necessary ingredient for the analysis of numerical methods for this problem based on finite elements or finite differences. [Copyright &y& Elsevier]

This paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure. The main requirement is that the agglomerates share faces that have closed boundaries composed of 1-d edges. The spaces resulting from the agglomerated elements are subspaces of the original de Rham sequence of H1-conforming, H(curl)-conforming, H(div)-conforming, and piecewise constant spaces associated with an unstructured "fine" mesh. The procedure can be recursively applied so that a sequence of nested de Rham complexes can be constructed. As an illustration we generate coarser spaces from the sequence corresponding to the lowest-order Nédélec spaces, lowest-order Raviart-Thomas spaces, and for piecewise linear H1-conforming spaces, all in three dimensions. The resulting V-cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case. [ABSTRACT FROM AUTHOR]

This paper analyses a two-level preconditioning scheme for H(curl) bilinear forms. The scheme utilizes an auxiliary problem on a related mesh that is more amenable for constructing optimal order multigrid methods. More specifically, we analyse the case when the auxiliary mesh only approximately covers the original domain. The latter assumption is important since it allows for easy construction of nested multilevel spaces on regular auxiliary meshes. Numerical experiments in both two and three space dimensions illustrate the optimal performance of the method. Published in 2007 by John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We present a negative-norm least-squares method for axisymmetric div–curl systems arising from Maxwell's equations for electrostatics and magnetostatics in three dimensions. The method approximates the solution in a two-dimensional meridian plane. To achieve this dimension reduction, we must work with weighted spaces in cylindrical co-ordinates. In this setting, a stable pair of approximation spaces is developed and analysed. We also report the results of some numerical experiments, which demonstrate a quasi-optimal convergence rate and a robustness with respect to the domain and coefficients. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we develop a multigrid preconditioner for the discrete system of linear equations that results from the mixed formulation of the linear plane elasticity problem using the Arnold-Winther elements. This, in turn, can be reduced to the problem of finding a multigrid preconditioner for the form (·,··) + (div·, div·) in the symmetric matrix space resulting from Arnold-Winther elements. Since the form is not uniformly elliptic, a Helmholtz-type decomposition is essential. The Arnold-Winther finite element space gives rise to nonnested multilevel spaces adding difficulty to the analysis. We prove that for the variable V-cycle multigrid preconditioner, the condition number of the preconditioned system is independent of the number of levels. The results of numerical experiments are also presented. [ABSTRACT FROM AUTHOR]

This paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic Maxwell equations. In particular, a "backslash" multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used. The method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator. This idea has previously been used in multigrid analysis of indefinite second order elliptic problems. However, the Maxwell application involves a nonelliptic indefinite operator. With the help of a few new estimates, the earlier ideas can still be applied. Some numerical experiments with lowest order Nedelec elements are also reported. [ABSTRACT FROM AUTHOR]

In this paper, we consider approximation of a second-order elliptic problem defined on a domain in two-dimensional Euclidean space. Partitioning the domain into two subdomains, we consider a technique proposed by Wieners and Wohlmuth [9] for coupling mixed finite element approximation on one subdomain with a standard finite element approximation on the other. In this paper, we study the iterative solution of the resulting linear system of equations. This system is symmetric and indefinite (of saddle-point type). The stability estimates for the discretization imply that the algebraic system can be preconditioned by a block diagonal operator involving a preconditioner for H(div) (on the mixed side) and one for the discrete Laplacian (on the finite element side). Alternatively, we provide iterative techniques based on domain decomposition. Utilizing subdomain solvers, the composite problem is reduced to a problem defined only on the interface between the two subdomains. We prove that the interface problem is symmetric, positive definite and well conditioned and hence can be effectively solved by a conjugate gradient iteration. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider the finite element approximation of fractional powers of regularly accretive operators via the Dunford–Taylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A. Bonito and J. E. Pasciak, IMA J. Numer. Anal., 37 (2016), No. 3, 1245–1273] by reducing the regularity required on the data. Numerical experiments illustrating the new theory are provided. [ABSTRACT FROM AUTHOR]

We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator L defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on H 0 1 ( Ω ) . The time dependent solution u ( x , t ) is represented as a Dunford–Taylor integral along a contour in the complex plane. The contour integrals are approximated using sinc quadratures. In the case of homogeneous right-hand-sides and initial value v , the approximation results in a linear combination of functions ( z q I − L ) − 1 v ∈ H 0 1 ( Ω ) for a finite number of quadrature points z q lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements. Our main result provides L 2 ( Ω ) error estimates between the solution u ( ⋅ , t ) and its final approximation. Numerical results illustrating the behavior of the algorithms are provided. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in and - norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results. [Copyright &y& Elsevier]

We consider the Laplace equation under mixed boundary conditions on a polygonal domain Ω. Regularity estimates in terms of Sobolev norms of fractional order for this type of problem are proved. The analysis is based on new interpolation results and multilevel representation of norms on the Sobolev spaces Hα(Ω). The Fourier transform and the construction of extension operators to Sobolev spaces on ℝ2 are avoided in the proofs of the interpolation theorems. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions on globally or locally quasi-uniform meshes. A set of abstract conditions is given for the multiplier spaces which are sufficient to guarantee a stable and convergent mortar approximation. Three examples of multipliers satisfying these conditions are presented. The first one is a dual basis example, while the remaining two are based on finite volumes. Finally, the results of computational examples illustrating the theory are reported. [ABSTRACT FROM AUTHOR]