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 H1 and L2 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.