Method of particular solutions for nonlinear Poisson-type equations in irregular domains

Abstract

Abstract The paper presents a new meshless numerical technique for solving nonlinear Poisson-type equations ∇ 2 u = F ( u , u x 1 , u x 2 , x ) + f ( x ) in irregular domains. Using the dual reciprocity method (DRM) approach, the nonlinear term is represented as a linear combination of basis functions F ≃ ∑ m M q m φ m . We use RBFs as the basis functions φ m . According to the DRM this admits of writing the approximate solution in the form of the linear combination of particular solutions corresponding to these basis functions u M = u f + ∑ m M q m Φ m . We use the particular solution as a sum of two parts Φ = Φ 0 + Φ 1 , where Φ 0 is the well known analytical solution of the Poissone equation with a RBF on the right-hand side. The second term Φ 1 provides that Φ satisfies the homogeneous conditions on the boundary of the domain. As a result, it is possible to isolate the nonlinear part of the equation and the problem is reduced to a system of nonlinear equations F ( u M , u M , x 1 , u M , x 2 , x ) = ∑ m M q m φ m for the unknown coefficients qm. Then the nonlinear system is solved numerically. Numerical experiments are carried out for accuracy and convergence investigations. A comparison of the numerical results obtained in the paper with the exact solutions or other numerical methods indicates that the proposed method is accurate and efficient in dealing with complicated geometry and strong nonlinearity.