Finite Volume Element Method For Nonlinear Equations On Quadrilateral Meshes

Nonlinear equations are widely used in natural science and social science.We can use it to simulate physical process and solve problems in ecological system and economic system.However,most nonlinear equations have no analytical solutions.So it is important to study its numerical methods.In this paper,we solve the second-order nonlinear elliptic equation by using the finite volume element method,and give the rigorous error estimates.Firstly,the solution region is divided into general convex quadrilateral meshes.We choose the isoparametric bilinear element space as the trial function space and the piecewise constant function space as the test function space.And we construct the corresponding finite volume element scheme.Secondly,on the h2-parallelogram grids,we give the regularity assumption of the element,and then deduce the boundedness and positive definiteness of bilinear form.Then,under the regularity condition of the exact solution u?H3,we divide the error estimates into two parts:H1 error and L2 error.H1-error estimate contains L2 error,and L2-error estimate contains H1 error.By using ?u-uh?0?0(h?0),we obtain the optimal error estimates of H1 and L2,whose convergence rates are O(h)and O(h2),respectively.We carry out numerical experiments on any quadrilateral meshes and calculate the optimal convergence orders under H1-norm and L2-norm.The numerical results are consistent with the theoretical results,which verifies the correctness of the theoretical analysis.