Numerical solution of nonlinear elliptic equation with nonlocal condition

Abstract

Two iterative methods are considered for the system of difference equations approximating two-dimensional nonlinear elliptic equation with the nonlocal integral condition. Motivation and possible applications of the problem present in the paper coincide with the small volume problems in hydrodynamics. The differential problem considered in the article is some generalization of the boundary value problem for minimal surface equation.

Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions

“…Systems 12- (14) and 18, (19), (16), (14) are equivalent. So 12- 14is being reduced to two separate systems of lower order: one system is (18), (19), (14) and the other system is (16). In system (18), (19), (14) only the unknowns z ij , i, j = 1, N -1, in the internal points of the domain D are presented.…”

“…So the number of equations and unknowns in the system is equal to (N -1) 2 . Furthermore, the unknowns z 0j in (16) are expressed by unknowns z ij , i, j = 1, N -1. This allows us to first solve (18), (19), (14) and 16only afterwards.…”

“…Various iterative methods for the systems of linear difference equations, derived from elliptic equations with nonlocal conditions, and proofs of convergence of these methods can be found in [13][14][15]. In the articles [16][17][18][19], the iterative methods are generalized for the systems of nonlinear difference equations with nonlocal conditions. Some other difference methods for elliptic equations with nonlocal conditions were described in [13,[20][21][22].…”

Well-posedness of a nonlocal boundary value difference elliptic problem

“…Such conditions usually are identified as nonlocal boundary conditions and reflect situations when the data on the domain boundary cannot be measured directly, or when the data on the boundary depend on the data inside the domain. The well-posedness of various nonlocal boundary value problems for partial differential and difference equations has been studied extensively by many researchers (see, e.g., [5][6][7][8][9][10][11][12][13][14][15][16][17][22][23][24] and the references given therein).…”

Application of M-matrices theory to numerical investigation of a nonlinear elliptic equation with an integral condition

“…Iterative methods for systems of nonlinear difference equations with nonlocal conditions were considered in [9,31] (also see paper [36] close to the subject area).…”