Superconvergent biquadratic finite volume element method for two-dimensional Poisson's equations

The authors consider the biquadratic finite volume element approximation for the Pois-son's equation on the rectangular domain Ω = (0, 1) 2. The primal mesh is performed using a ractangular partition. The control volumes are chosen in such a way that the vertices are stress points of the primal mesh. In order to solve the scheme more efficiently, the authors wrote the bi-quadratic finite volume element scheme as a tensor product form and used the alternating direction technique to solve it. Thanks to the fact that the primal mesh satisfies a superconvergence property in the interpo-latory approximation, the authors prove that the numerical gradients of the method have h 3 – superconvergence order at optimal stress points. Using the dual argument technique, the authors also prove that the convergence order in L 2 –norm is h 4 at nodal points. A numerical example is presented to support the theoretical results. Subject classification: 65N12; 65N15 To be checken if really these subject classification are those of 2010 or not 1 Basic knowledge and motivation 1. (definition): Finite volume element methods, biefly FVEM (called also box methods in its early time and generalized difference methods in China) discretize integral form of conservation law of differential equations by chosing linear or bilinear finite element spaces as trial spaces. 2. (uses...): FVEM have been widely used in the numerical approximation of partial differential equations because they keep the conservation law of mass or energy. 3. (interpolation...): both finite element and finite volume element methods are both based on the interpolations: