A virtual element constructed based on a class of quasi-orthogonal polynomials and its application to fourth-order elliptic problems

In order to improve an existing classical virtual element for the fourth-order elliptic problem, we propose a special class of quasi-orthogonal polynomials and then construct a new type of numerical integration formula based on the polynomials. The special feature of the formula is that it uses the function values and the derivative values of the integrand at the endpoint of the interval, which is of some practicality in virtual element construction and other occasions. Then, we replace the edge integral degrees of freedom (DoFs) in the existing virtual element with the function values at the integral points of the above formula, and propose a modified virtual element for solving the fourth-order elliptic problem. We analyze the convergence and computability of the modified virtual element, and the results show that the element has the characteristics of simpler programming implementation and lower computational complexity while maintaining the original convergence order. Finally, numerical results are given to verify the convergence of the element in solving the fourth-order elliptic problem.