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In this paper, we develop and study algorithms for approximately solving linear algebraic systems: |${{\mathcal{A}}}_h^\alpha u_h = f_h$|⁠ , |$ 0< \alpha <1$|⁠ , for |$u_h, f_h \in V_h$| with |$V_h$| a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem |$ {{\mathcal{A}}}^\alpha u=f$| with |${{\mathcal{A}}}$|⁠ , for example, coming from a second-order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the method of Vabishchevich (2015, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys. , 282 , 289–302) that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval |$[0,1]$|⁠. Here we develop and study two time-stepping schemes based on diagonal Padé approximation to |$(1+x)^{-\alpha }$|⁠. The first one uses geometrically graded meshes in order to compensate for the singular behaviour of the solution for |$t$| close to |$0$|⁠. The second algorithm uses uniform time stepping, but requires smoothness of the data |$f_h$| in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of |$f_h$| playing a major role for the second method. Finally, we present numerical experiments for |${{\mathcal{A}}}_h$| coming from the finite element approximations of second-order elliptic boundary value problems in one and two spatial dimensions. [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]

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]

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]