Localized boundary-domain integral formulations for problems with variable coefficients

Abstract

Specially constructed localized parametrixes are used in this paper instead of a fundamental solution to reduce a Boundary Value Problem with variable coefficients to a Localized Boundary-Domain Integral or Integro-Differential Equation (LBDIE or LBDIDE). After discretization, this results in a sparsely populated system of linear algebraic equations, which can be solved by well-known efficient methods. This make the method competitive with the Finite Element Method for such problems. Some methods of the parametrix localization are discussed and the corresponding LBDIEs and LBDIDEs are introduced. Both mesh-based and meshless algorithms for the localized equations discretization are described.

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

“…Alternatively, the normal derivative of u on Γ can be computed numerically and the result is an integro-differential equation on Ω only. The same possibilities hold if (5) is chosen [27], the difference being that the kernel of the area integral now involves ∇ x G(x, y).…”

“…We set the requested tolerance very high, often to 5 · 10 −16 , as to test the stability of our code. In a typical example of a matrix-vector multiplication we first evaluate M 0 u∆r(x) using the fast multipole method and the formula (27) for all source and target points irrespective of whether they are close to each other or not. Then we add the sum of all integrals (28) minus a correction, which is the part of (27) which should not have been included.…”

“…Simkin and Trowbridge [34], Mayo [25], and Mikhailov [27] all discuss integral equations which, in a sense described in Section 3, are related to the splitting (5). Neither Mayo [25] nor Mikhailov [27] do actually solve any equations.…”

“…Neither Mayo [25] nor Mikhailov [27] do actually solve any equations. Simkin and Trowbridge [34] only use integral equations to solve problems with piece-wise constant σ.…”

“…See, for example, [27,34] for explicit expressions in 2D and 3D, and [25] for an example involving an unbounded region Ω. Let us briefly examine the integral equations derived in the manner just described.…”