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We describe and analyze the weighted extended b-spline (WEB-Spline) mesh-free finite element method for solving the p-biharmonic problem. The WEB-Spline method uses weighted extended b-splines as basis functions on regular grids and does not require any mesh generation which eliminates a difficult, time consuming preprocessing step. Accurate approximations are possible with relatively low-dimensional subspaces. We perform some numerical experiments to demonstrate the efficiency of the WEB-Spline method.

In this paper, we consider a biharmonic equation with Navier boundary conditions and we decoupled this fourth order biharmonic equation into a system of second order equations. In numerics, there is an advantage because of this decoupling, i.e., one can use C 0 finite elements to solve the decoupled system instead of C 1 finite elements. For solving this decoupled system of second order equations, we use the weighted extended b-spline (WEB-Spline) based mesh-free finite element method. The WEB-Spline method does not require any mesh generation and eliminates the difficult, time-consuming preprocessing step. Also, by the WEB-Spline method higher order approximations are possible with relatively low dimensional spaces. Numerical results based on the WEB-Spline method are compared with the usual finite element method in order to demonstrate the efficiency of the proposed method.

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