Abstract
A meshless method for the numerical solution of partial differential equations with nonlinear inequality constraints is discussed in this paper. The original nonlinear inequality problem is linearized as a sequence of linear equality problems, and then discrete linear system of algebraic equations is formed. This meshless method only requires nodes on the boundary of the domain, and it does not require any numerical integrations. Numerical experiments indicate that this method is very effective for nonlinear inequality problems and has good convergence rate and high computational efficiency.
Highlights
This paper concerns numerical solutions of the following elliptic boundary value problem with nonlinear inequality constraints: u =, in, ( ) u = u, on D, q = q, on N,u ≤ f, q ≤ g, (u – f )(q – g) =, on R, where is a plane domain bounded by the boundary, u is an unknown function, q := ∂u/∂n is the normal derivative of u, n is the outward normal to, uis the given potential on D, qis the given normal flux on N, and f and g are prescribed functions on R =– D – N
The aim of this paper is to develop a boundary-type meshless method for the numerical solution of the nonlinear inequality problem ( )-( )
Summarizing, we have shown that the nonlinear inequality constraint ( ) is equivalent to Eq ( )
Summary
This paper concerns numerical solutions of the following elliptic boundary value problem with nonlinear inequality constraints:. The aim of this paper is to develop a boundary-type meshless method for the numerical solution of the nonlinear inequality problem ( )-( ). This method only requires boundary nodes, and does not need any mesh for either interpolation or integration. When boundary nodes are used on each side of the domain, Figure depicts numerical results of our method and the BEM [ ] for u on R. Figure gives the numerical results of our method and the BEM [ ] when boundary nodes are used. We can see that the results are indistinguishable from those in [ , ]
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