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We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (1) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (2) truncate each elliptic problem to a bounded domain, (3) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter. [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]

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e. an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator E(t) for the initial value problem can be written as a Dunford-Taylor integral involving the Mittag-Leffler function e α,1 and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here α denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator W(t) and the forcing function F(t). We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford-Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of 1/h . The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying a sinc quadrature technique to approximate the Dunford-Taylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size k involves k-2 nodes and results in an additional O(exp(-c/k)) error. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo-midpoint quadrature. This involves the average of Wh(s), (the semi-discrete approximation to W(s)) over the quadrature interval. This average can also be written as a Dunford-Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford-Taylor integrals for computing the averages. We show that for a refined grid in time with a mesh of O(Nlog(N)) intervals, the error between the semi-discrete and fully discrete approximation is O(N-2 + log(N) exp(-c/k )). We also report the results of numerical experiments that are in agreement with the theoretical error estimates. [ABSTRACT FROM AUTHOR]

We study the numerical approximation of fractional powers of accretive operators in this article. Namely, if A is the accretive operator associated with a regular sesquilinear form A(.,.) defined on a Hilbert space V contained in L2(Ω), we approximate A-β for β ∊(0, 1). The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space Vh ⊂ V, A-β is approximated by A-β h πh, where Ah is the operator associated with the form A(.,.) restricted to Vh, and πh is the L2(Ω)-projection onto Vh. We first provide error estimates for (A-β - A-β h πh)f in Sobolev norms with index in [0,1] for appropriate f . These results depend on the elliptic regularity properties of variational solutions involving the form A(.,.) and are valid for the case of less than full elliptic regularity. We also construct and analyse an exponentially convergent sinc quadrature approximation to the Balakrishnan integral defining A-β h πhf . Finally, the results of numerical computations illustrating the proposed method are given. [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]

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 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]

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]

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]

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]