Solution of double-sided boundary value problems for the Laplacian in R³ by means of potential theory methods

About Three Dimensional Jump Boundary Value Problems for the Laplacian

“…Using formula (3) and conditions (21) we have the equivalent to (9), (21) variational problem: to find the function . Then from Lax-Milgram theorem [16] we obtain [17] Theorem 12. The problem (9), (21) has one and only one solution.…”

“…These results allow us to use projection methods [7,8] for numerical solution of such integral equations, avoiding the use of resource-consuming regularization procedures [9]. The need to determine the conditions of well-posed solvability also arises when the sum of simple and double layer potentials is used to solve the double-sided Dirichlet and Neumann problems [10] or doublesided Dirichlet-Neumann problem [11] in the space of functions that, same as their normal derivatives, have a jump on crossing boundary surface. When the domain and its environment have different physical properties, there is a need to solve the boundary value problems with jump conditions.…”

Finite element approximations in projection methods for solution of some Fredholm integral equation of the first kind

“…Assume that surface  satisfy the conditions of p. . Define analogously to (13), (14) in each grid domain…”

“…When solving Neumann problem in the space of functions with jump on crossing boundary surface using double layer potential, we also proceed to integral equation of the first kind [11,12]. The need to solve integral equations of the first kind also arises when the sum of simple and double layer potentials is used to solve the double-sided Dirichlet or Neumann problem [13] or double-sided Dirichlet-Neumann problem [14] in the space of functions that, same as their normal derivatives, have jump on crossing boundary surface. Many systems of integral equations for the simple and double layer potentials that are equivalent to mixed boundary value problems for Laplace equation, also contain integral equations of the first kind [15,16].…”

“…In general, researches of projection methods convergence mainly focus on solving integral equations of the second kind [4,6,17]. Defining well-posed solvability conditions for integral equations of the first kind that are equivalent to boundary value problems for Laplace's equation in Hilbert spaces [18][19][20] allows us to use projection methods for numerical solution of such equations, thus avoiding resource-consuming regularization procedures [21][22][23]. For detailed review of numerical methods for solving integral equations, please see [2][3][4]6].…”