Efficient Finite Element Method Based On The Dimension Reduction Scheme For Fourth Order Partial Differential Equation

In this paper,the effective finite element methods based on the dimension reduction scheme for the fourth order equations in the circular/spherical domain are respectively proposed.For the circular domain,by using polar coordinate transformation and Fourier basis function expansion,the original problem is transformed into a series of equivalent one-dimensional fourth-order problems.Then,we introduce appropriate weighted Sobolev space and establish the weak form and corresponding discrete scheme for each fourth-order problem.The error estimates of approximation solutions are proved by using the approximation properties of cubic Hermite interpolation operator.In addition,we construct the cubic Hermite interpolation basis functions to translate discrete scheme into the corresponding matrix form.Finally,we give some numerical examples,and the numerical results indicate that the effectiveness of our algorithm.For the spherical domain,in order to overcome the pole singularity introduced by spherical coordinate transformation,we derive the essential pole condition.Using the orthogonality of spherical harmonic function,the original problem is transformed into a series of equivalent one-dimensional fourth-order problems.Then according to the polar condition,an appropriate weighted Sobolev space is introduced,and the weak form of each one-dimensional fourth-order problem and the corresponding discrete scheme are derived.By using the Lax-milgram theorem and the properties of the cubic Hermite interpolation operator,we prove the existence and uniqueness of the weak and numerical solutions and the error estimates of the approximate solutions.Finally,we give some numerical examples,and the numerical results also show the effectiveness of our algorithm.