A Comparison of Splittings and Integral Equation Solvers for a Nonseparable Elliptic Equation

Abstract

Iterative numerical schemes for solving the electrostatic partial differential equation with variable conductivity, using fast and high-order accurate direct methods for preconditioning, are compared. Two integral method schemes of this type, based on previously suggested splittings of the equation, are proposed, analyzed, and implemented. The schemes are tested for large problems on a square. Particular emphasis is paid to convergence as a function of geometric complexity in the conductivity. Differences in performance of the schemes are predicted and observed in a striking manner.

Norm-Preserving Discretization of Integral Equations for Elliptic PDEs with Internal Layers I: The One-Dimensional Case

“…We see from the data that the condition numbers of the discrete operators do, indeed, exhibit the scaling properties expected from our analysis of the continuous operators. Note that the 1norm-preserving scheme to discretize (7) and the ∞-norm-preserving scheme to discretize (10) result in very well-conditioned matrices, independent of the steepness of the internal layer.…”

“…This problem can be addressed using one of the integral equations ( 7) or (10), from which the solution to the original problem is u = v + l. Here, we consider f ≡ 1, γ a = 1 and γ b = 2. We consider two types of functions (x) that contain multiple internal layers by adding together several hyperbolic tangent functions, as in (25), with multiple centers and δ = 500, as shown in Fig.…”

“…Volume integral equations can also been used for problems such as (1). There is a substantial literature in this area, which we do not attempt to review, except to observe that there are a variety of analytic methods which can be used to derive integral formulations, a variety of numerical methods which can be used for their discretization, and a variety of fast algorithms which can be used for iterative or direct solution [5,6,8,9,10,11,16,18,20,26].…”

Quasi‐analytic models for density bubbles and plasma clouds in the equatorial ionosphere: Closed form solutions for electric fields and potentials

“…The numerical solution of nonseparable elliptic equations illustrated by has been discussed by many authors (see review by Englund and Helsing [2004]). Numerical solutions to this type of equation are influenced by (1) the finite difference approximations to the derivatives, (2) the boundary conditions, and (3) the convergence error for iterative solvers.…”

A Nyström scheme with rational quadrature applied to edge crack problems

“…Yet another way to handle non-polynomial integrands is represented by the generalized Gaussian quadrature rules presented by Yarvin and Rokhlin [6]. For an example where such rules are used, see [7]. Being Gaussian, these quadrature rules integrate 2k basis functions exactly using k quadrature nodes.…”