Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach

A computer-assisted existence proof for Emden's equation on an unbounded L-shaped domain

We prove existence, non-degeneracy, and exponential decay at infinity of a non-trivial solution to Emden's equation $-\Delta u = | u |^3$ on an unbounded $L$-shaped domain, subject to Dirichlet boundary conditions. Besides the direct value of this result, we also regard this solution as a building block for solutions on expanding bounded domains with corners, to be established in future work. Our proof makes heavy use of computer assistance: Starting from a numerical approximate solution, we use a fixed-point argument to prove existence of a near-by exact solution. The eigenvalue bounds established in the course of this proof also imply non-degeneracy of the solution

Modeling elastic wave propagation in fluid-filled boreholes drilled in nonhomogeneous media: BEM – MLPG versus BEM-FEM coupling

The efficiency of two coupling formulations, the boundary element method (BEM)-meshless local Petrov–Galerkin (MLPG) versus the BEM-finite element method (FEM), used to simulate the elastic wave propagation in fluid-filled boreholes generated by a blast load, is compared. The longitudinal geometry is assumed to be invariant in the axial direction (2.5D formulation). The material properties in the vicinity of the borehole are assumed to be nonhomogeneous as a result of the construction process and the ageing of the material. In both models, the BEM is used to tackle the propagation within the fluid domain inside the borehole and the unbounded homogeneous domain. The MLPG and the FEM are used to simulate the confined, damaged, nonhomogeneous, surrounding borehole, thus utilizing the advantages of these methods in modeling nonhomogeneous bounded media. In both numerical techniques the coupling is accomplished directly at the nodal points located at the common interfaces. Continuity of stresses and displacements is imposed at the solid–solid interface, while continuity of normal stresses and displacements and null shear stress are prescribed at the fluid–solid interface. The performance of each coupled BEM-MLPG and BEM-FEM approach is determined using referenced results provided by an analytical solution developed for a circular multi-layered subdomain. The comparison of the coupled techniques is evaluated for different excitation frequencies, axial wavenumbers and degrees of freedom (nodal points).Ministerio de Economía y Competitividad BIA2013-43085-PCentro Informático Científico de Andalucía (CICA

Exact Boundary Conditions at an Artificial Boundary for Partial Differential Equations in Cylinders

The numerical solution of partial differential equations in unbounded domains requires a finite computational domain. Often one obtains a finite domain by introducing an artificial boundary and imposing boundary conditions there. This paper derives exact boundary conditions at an artificial boundary for partial differential equations in cylinders. An abstract theory is developed to analyze the general linear problem. Solvability requirements and estimates of the solution of the resulting finite problem are obtained by use of the notions of exponential and ordinary dichotomies. Useful representations of the boundary conditions are derived using separation of variables for problems with constant tails. The constant tail results are extended to problems whose coefficients obtain limits at infinity by use of an abstract perturbation theory. The perturbation theory approach is also applied to a class of nonlinear problems. General asymptotic formulas for the boundary conditions are derived and displayed in detail

Spectral discretization errors in filtered subspace iteration

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates