A new meshless method for solving steady-state nonlinear heat conduction problems in arbitrary plane domain

For solving steady-state nonlinear heat conduction problems (HCPs) in arbitrary plane domain enclosed by a complex boundary shape, the meshless methods are convenient to programming, of which the polynomial expansion method seems the simplest one. However, it is seldom used as a major medium to solve nonlinear partial differential equations owing to its highly ill-conditioned behavior. First, we decompose the steady-state nonlinear heat conduction equation into a linear portion on the left-hand side, and leave other terms on the right-hand side as a non-homogeneous portion. Then, we need to solve a steady-state linear heat conduction problem in each iteration, for which we propose a multiple-scale Pascal triangle method to generate the linear algebraic equations, where the multiple scales are automatically decided by the collocation points. We can find that these scales can largely reduce the condition number of the coefficient matrix in each linear system, such that the iteration process is convergent very quickly, and the numerical solutions obtained are very accurate and stable against very large noise to over 50%, no matter for the Dirichlet and mixed-type boundary value problems. Numerical results confirm the validity of the present multiple-scale polynomial expansion method for solving steady-state nonlinear HCPs in arbitrary plane domains.