Finite volume element method for nonlinear elliptic equations on quadrilateral meshes

In this paper, we solve a second-order nonlinear elliptic equation by using the finite volume element method, and give the rigorous error estimates. Firstly, the computational domain 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 construct the corresponding finite volume element scheme. Secondly, on the h(2)-parallelogram mesh, the boundedness and coercivity of bilinear form are proved. Using the Brouwer fixed point theorem, we give the existence and uniqueness of numerical solution. Thirdly, we derive the estimates of parallel to(del(u-u(h))parallel to with t >= 2 and parallel to u-u(h)parallel to(0) under certain regularity assumptions. At last, we carry out numerical experiments on quadrilateral meshes and calculate the convergence orders in H-1 and L-2 norms, which are consistent with our theoretical results.