A finite volume method on general surfaces and its error estimates

In this paper, we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface. The discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons. The optimal error estimates of the approximate solution are proved in both the H 1 and L 2 norms which are of first order and second order respectively under mesh regularity assumptions. Some numerical tests are also carried out to experimentally verify our theoretical analysis.