Nonlinear elliptic problems with the method of finite volumes

Abstract

We present a finite volume discretization of the nonlinear elliptic problems. The discretization results in a nonlinear algebraic system of equations. A Newton-Krylov algorithm is also presented for solving the system of nonlinear algebraic equations. Numerically solving nonlinear partial differential equations consists of discretizing the nonlinear partial differential equation and then solving the formed nonlinear system of equations. We demonstrate the convergence of the discretization scheme and also the convergence of the Newton solver through a variety of practical numerical examples.

Linearized Initialization of the Newton Krylov Algorithm for Nonlinear Elliptic Problems

“…Each cell in the mesh produces a nonlinear algebraic equation [15], [12]. Thus, discretization of the equations (1), (2) and (3) on a mesh with n cells result in n nonlinear equations, and let these equations are given as…”

“…For computing the true error and convergence behavior of the methods, let us further assume that the exact solution of the equations (7) and (8) Figure 1 displays the surface plot of the exact solution. We are discretizing equations (7) and (8) on a 40 × 40 mesh by the method of Finite Volumes [11], [12], [13], [15]. Discretization results in a nonlinear algebraic vector (4) with 1600 nonlinear equations.…”

“…But, still there is no optimal nonlinear solver, or the one that we know of. Our research is in the field of optimal solution of nonlinear equations generated by the discretization of the nonlinear elliptic equations [15], [14], [13], [12]. Let us consider the following nonlinear elliptic partial differential equation [15] div(−K grad p) + f (p) = s (x, y) in Ω (1)…”

“…These methods convert nonlinear partial differential equations into a system of algebraic equations. We are using the Newton Krylov algorithm for solving the discrete nonlinear system of equations formed by the Finite Volume method [15]. Since, initial guess or initialization is very important for the convergence of the Newton's algorithm.…”

Stopping Criterion for Adaptive Algorithm

“…The coefficient K is allowed to be discontinuous in space. In porous media flow [7,4,1], the unknown function p = p(x, y) represents the pressure of a single phase, K is the permeability or hydraulic conductivity of the porous medium, and the velocity u of the phase is given by the Darcy law as : u = −K grad p. The next section presents finite volume method and adaptive algorithm.…”