A Posteriori Error Estimation Of Rotating Bilinear Finite Volume Method For Elliptic Equations

In this paper,we discuss the posterior error estimation and adaptive algorithm of the non-conforming element finite volume method.Firstly,the posterior error estimation for the rotating bilinear elements finite volume method on the second-order elliptic equation is studied,and then we realize the adaptive algorithm of the triangular linear non-conforming element finite volume method.Consider the second-order elliptic boundary value problem (?) rectangular division_his performed on the area?,take the trial function space V_h^ncas the rotating bilinear element space.By connecting two diagonals of each rectangular unit,we obtained the dual division (?)_h^*,and take the test space V_h^* as the piecewise constant function space corresponding to (?)_h^*.On this basis,the finite volume method of rotating bilinear element is defined.The H¹ and L² convergence order estimation of this format has been discussed in literature.In this article,we focus on the posterior error estimation.At first,the posterior error indicator based on H¹ mode is defined,after that,we prove the equivalence between the posterior error and prior error.Furthermore,we define the L² mode posterior error indicator,and the equivalence of the posterior error and the prior L² error is proved.Finally,the relationships between two posterior errors and their corresponding prior errors are verified by numerical experiments.Our another target is to achieve adaptive algorithm.Since the adaptive algorithm on the rotating bilinear element finite volume method will appear hanging points when dividing mesh,this phenomenon can be avoided on the triangular mesh.Therefore,we implement the adaptive algorithm on the triangular linear non-conforming element finite volume method.The implementation of the adaptive algorithm is divided into the following steps.First of all,we performe the numerical solution on the initial grid,select the L² posterior error indicator as an reference of the size of the error on each unit.Afterwards,we can obtain a new mesh by dividing the unit with a large error,recalculate the numerical solution,and repeat the above steps until the error is satisfactory.Through the adaptive numerical example,we can find that compared with the finite volume method under a uniform mesh,the adaptive algorithm significantly improves the computational efficiency.