Generalized Finite Integration Method with Volterra operator for multi-dimensional biharmonic equations

In this paper, we propose an improved Generalize Finite Integration Method with Volterra operator to solve multi-dimensional biharmonic equations. The novelty of the proposed method is that its mth order integration matrix will be exactly formulated by using Volterra operator. This leads to a significant improvement in the accuracy of the approximation. Furthermore, the original strict requirement of uniformly distributed collocation points is waived in the proposed method due to the use of piecewise polynomials. This contributes to another distinct advantage in placing collocation points for problems with geometrically complex domains. Four numerical examples are constructed to demonstrate these advantages. Comparisons on the accuracies obtained from quadrature formulas with and without Volterra operator are given. Numerical results indicate that the proposed method is capable for solving multi-dimensional biharmonic equations with superior efficiency, higher accuracy and fewer memory usage.