The Finite Volume Element Method For The Poisson Equation Based On Adini's Element

Let us consider the following Poisson problem, (0-1) whereThe trial function Space Uh is Adini's element, and the test function Space Vh is the piecewise constant function,the following finite volume scheme is:Find uh∈Uh, such that (0-2) which is equivalent to, (0-3)Thanks to Green formula,we can have (0-4) (0-5) (0-6)According to the definition of Adini's element,in K (0-7)By substituting the expression of uh in each control volume K to the equa-tions(0-4),(0-5),(0-6),we can get the difference scheme from the integration over each control volume K.We can prove a(uh,π*uh) is positive definite by cell analysis method.We can make the matrix A symmetric by setting A= (A+AT)/2, and the eigenvalues of the new matrix A are all positive, Since the matrix A is positive definite,that is to say a(uh,πh*uh) is positive definite.Hence,there exists a constant number a> 0,independent of Uh such that (0-8)That is to say: Theorem 0.1 We assume that the rectangular meshΓh* satisfies the boundary condition.Let u,uh be the exact solution of (0-1) and the finite element solution of (0-2). If u∈H4(Ω), we can get (0-9)In order to get the error estimate of H1 norm,we need the definition of the discrete norm, (0-10)We notice that in the HE1 space,the semi-norm |·|1 is equivalent to ||·||1, hence we just need compute |·|1 norm.The results of |·|1 can be obtained by integrating (0-10)Theorem 0.2 We assume that the rectangular meshΓh* satisfies the boundary condition.Let u,uh be the exact solution of (0-1) and the finite element solution of (0-2). If u∈H4(Ω), we can get the following error estimate: (0-11)We will present and prove the error estimate of L2, we introduce the L2 norm, (0-12)We can get the error estimate of L2:Theorem 0.3 We assume that the rectangular meshΓh* satisfies the boundary condition.Let u,uh be the exact solution of (0-1) and the finite element solution of (0-2). If u∈H4(Ω), we can get the following error estimate: (0-13)The error estimate of the superconvergence can also be given. Due to Th6.1 in chapter Two of《The General Difference Method》,we will know that:Lemma 0.1 Thus,We also can prove this:Theorem 0.4 We assume that the rectangular meshΓh* satisfies the boundary condition.Let u,uh be the exact solution of (0-1) and the finite element solution of (0-2). If u∈H4(Ω), we can get the following error estimate: (0-14)Finally,we present the practical implementation of the finite volume el-ement method.We also give the numerical results.We can get linear equa-tions, we can solve the equations by Matlab.We can define the accurate solution of the elliptic equation as follows: (0-15)We get the errors of the function's value and the partial derivative's value. We get that the order of the convergence of H1, L2,the superconvergence,the Maximum norm is 3,4,4,4 respectively for both FVM and FEM.