Calculations Of Morley Finite Element On Arbitrary Quadrilateral Meshes

The fourth-order ellipse problem is widely used in the branches of elasticity and fluid mechanics.Therefore,it is very meaningful to study the numerical solution of the fourth-order ellipse problem.In theory,the finite element of the fourth-order problem needs to satisfy that the shape function space is a subspace ofH~2(?),then the function in the finite element space needs to be class of C~1 and piecewise smooth.It is very difficult to construct a function that satisfies the above conditions.Nonconforming element is a more commonly used finite element method,and its appearance reduces the requirement for function continuity.In practice,a simpler geometric shape is generally selected to make the meshes.Relatively speaking,it is more difficult to calculate on arbitrary quadrilateral mesh.In this paper we propose a new algorithm to evaluate the basis functions of the Morley finite element and their derivatives.The algorithm gives an efficient way to calculate the matrix which gives the change of coefficients between the bases for the reference and for an arbitrary quadrilateral.This matrix is factored as the product of two sparse matrices with a strong block structure.The new algorithm keeps the original accuracy,but the amount of calculation is reduced and the time is less.Numerical experiments verify our theoretical results.