On Certain Hybrid Iterative Methods for Solving Boundary Value Problems

Abstract

A mathematical justification of certain new iterative schemes used in solving the Dirichlet and Neumann problems for the Laplace equation is given. These schemes are based on a combination of Green's formula and some numerical method, FEM, for example, which is applied to some auxiliary mixed type boundary value problem on a subset of the original domain.In this paper we study some iterative methods used in solving the Dirichlet and Neumann problems for the Laplace equation. Each iteration is reduced to a calculation of harmonic potentials and to a subsequent solution of the mixed boundary value problem for the Laplace equation in a boundary strip. Both internal and external boundary value problems are considered. Under certain assumptions on the geometry of the surface we prove the convergence of successive approximations to the solution with the rate of geometrical progression.The numerical performance of this iterative scheme was given in [3] without mathematical justification. The mixed problem in a boundary strip was solved by the finite element method. It turned out that in some cases the iterative process is more effective than the finite element method for the whole domain. This is due to the comparatively low order of the system of algebraic equations one has to solve in each step of the iterative process and due to the possibility of using only the backward Gauss procedure for their solution.

A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain

“…Then Theorem 3.16 in [12] guarantees that the corresponding single-layer boundary integral operator S ii ϕ i is injective. Consider the following homogeneous problem:…”

“…In the past years, some numerical methods, such as the iterative boundary element method [8], the Tikhonov regularization approach [9], including the alternating iterative method [10][11][12], the conjugate gradient method [13], energy regularization method [14], some other regularization method [15], have been proposed to deal with the Cauchy problem for the Laplace equation [8,16]. Although many regularization methods have been applied to solve the Cauchy problem for the Laplace equation in annulus domain [14,16], we note that there are much fewer works devoted to the Cauchy problem of the Laplace equation in a general doubly connected planar domain [10,17,18] instead of an annular domain between concentric circles, but such a problem has important physical applications in the engineering field.…”

Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

“…In [34], a mathematical justification of certain new iterative schemes used in solving the Dirichlet and Neumann problems for the Laplace equation is given. These schemes are based on a combination of Green's formula and some numerical methods, FEM, for example, which is applied to some auxiliary mixed boundary value problem on a subset of the original domain.…”

“…These schemes are based on a combination of Green's formula and some numerical methods, FEM, for example, which is applied to some auxiliary mixed boundary value problem on a subset of the original domain. The proofs in [34] rely upon geometrical requirements on the boundary of the domain of strong convexity type which do not seem natural. [34].…”

“…The proofs in [34] rely upon geometrical requirements on the boundary of the domain of strong convexity type which do not seem natural. [34].…”