Polynomial-based mean weighted residuals methods for elliptic problems with nonlocal boundary conditions in the rectangle

Abstract

Abstract. In this paper, polynomial-based mean weighted residuals methods for the solution of elliptic problems with nonlocal boundary conditions, in rectangular domains, are numerically studied. When these methods are employed, the nonclassical boundary conditions involve the solution of large systems of linear equations or least squares problems, hence some numerical techniques for these solvers are compared to show the importance of using efficient algorithms for this purpose. Different kinds of nodes are used to demonstrate how they can be employed to solve different numerical problems when large derivatives of the solution appear. We will also study how using extra precision and/or oversampling can often reduce the computational effort.These methods can also be combined with others (as, for example, finite difference or spline methods) to solve linear and nonlinear parabolic or hyperbolic partial differential equations. The numerical study of some techniques as those explained above can help to obtain significantly better numerical approximations with a smaller computational effort.

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

“…Conclusion 2. If 0 γ < γ 0 and hypothesis (H2) is true, then iterative methods (16), (17), (18), (20) and (21) for system (4)-(6) expressed by form (12) converge. Conclusion 1 is valid for these iterative methods.…”

“…Other numerical methods for elliptic equations with various types of nonlocal conditions were explored in [17,19,21,22,35] (see also the references therein).…”

A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

“…Martín-Vaquero numerically study polynomial-based mean weighted residuals methods for the solution of elliptic problems with NBCs in rectangular domains [133].…”

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

“…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].…”