A FINITE ELEMENT COLLOCATION METHOD WITH SMOOTHED NODAL GRADIENTS

The collocation formulation has the salient advantages of simplicity and efficiency, but it requires the employment of high order gradients of shape functions associated with certain discretized strategies. The conventional finite element shape functions are usually C^0 continuous and thus cannot be directly adopted for the collocation analysis. This work presents a finite element collocation method through introducing a set of smoothed gradients of finite element shape functions. In the proposed formulation, the first order nodal smoothed gradients of finite element shape functions are defined with the aid of the general gradient smoothing methodology. Subsequently, the first order smoothed gradients of finite element shape functions are realized by selecting the finite element shape functions as the kernel functions for gradient smoothing. A further differential operation on the first order smoothed gradients then leads to the desired second order smoothed gradients of finite element shape functions, where it is noted that the conventional first order gradients are replaced by the first order smoothed gradients of finite element shape functions. It is theoretically proven that the proposed smoothed gradients of linear finite element shape functions not only meet the first order gradient reproducing conditions that are also satisfied by the conventional gradients of finite element shape functions, but also meet the second order gradient reproducing conditions for uniform meshes that cannot be fulfilled by the conventional finite element formulation. The proposed smoothed gradients of finite element shape functions enable a second order accurate finite element collocation formalism regarding both L_2 and H_1 errors, which is one order higher than the conventional linear finite element method in term of H_1 error, i.e., a superconvergence is achieved by the proposed finite element collocation method with smoothed nodal gradients. Numerical results well demonstrate the convergence and accuracy of the proposed finite element collocation method with smoothed nodal gradients, particularly the superior convergence and accuracy over the conventional finite element method according to the H_1 or energy errors.