On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients

In this paper, we propose a novel non-standard Local Fourier Analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the prediction capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis helps us to estimate a-priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method

Semidirect computation of three-dimensional viscous flows over suction holes in laminar flow control surfaces

A summary is given of the attempts made to apply semidirect methods to the calculation of three-dimensional viscous flows over suction holes in laminar flow control surfaces. The attempts were all unsuccessful, due to either (1) lack of resolution capability, (2) lack of computer efficiency, or (3) instability

Determination of normalized electric eigenfields in microwave cavities with sharp edges

The magnetic field integral equation for axially symmetric cavities with perfectly conducting piecewise smooth surfaces is discretized according to a high-order convergent Fourier--Nystr\"om scheme. The resulting solver is used to accurately determine eigenwavenumbers and normalized electric eigenfields in the entire computational domain.Comment: 34 pages, 6 figure

Spectral Solution with a Subtraction Method to Improve Accuracy

This work addresses the solution to a Dirichlet boundary value problem for the Poisson equation in 1-D, d2u/dx2 = f using a numerical Fourier collocation approach. The order of accuracy of this approach can be increased by modifying f so the periodic extension of the right hand side is suffciently smooth. A proof for the order is given by Sköllermo. This work introduces a subtraction technique to modify the function\u27s right hand side to reduce the discontinuities or improve the smoothness of its periodic extension. This subtraction technique consists of cosine polynomials found by using boundary derivatives. We subtract cosine polynomials to match boundary values and derivatives of f. The derivatives need only be calculated numerically and approximately represent derivatives at the boundaries. Increasing the number of cosine polynomials in the subtraction technique increases the order of accuracy of the solution. The use of cosine polynomials matches well with the Fourier transform approach and is computationally efficient. Implementation of this technique results in a solution with variable accuracy depending on the number of collocation points and approximated boundary derivatives. Results show that the technique can be up to 14th order accurate

Fast integral equation methods for the modified Helmholtz equation

We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) or $O(N\log N)$ operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of the methods on several numerical examples.Comment: Published in Computers & Mathematics with Application

Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension

We present a fast direct solver for two dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of the Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require $O(N^{3/2})$ work, where $N$ denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both the low and high frequency regimes.Comment: 18 page