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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]

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]

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]

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]