A quadratic finite volume method for nonlinear elliptic problems (Q2027790)

The authors consider a high order finite volume finite element method to approximate a large class of noninear elliptic equations in two dimensions. The numerical method uses the quadratic finite element spaces on the primal mesh. A nonlinear scheme is established. The existence and uniqueness of the discrete solution is proved. An error estimate of order \(O(h|\ln h|)\) in the \(W^{1,\infty}\)-norm is derived. It is also shown that the discrete solution has an optimal error estimate of order \(O(h^2)\) in the \(H^{1}\)-norm. Some numerical tests are presented to confirm the theoretical results. These error estimates are proved under some suitable assumptions on the exact solution.