ON THE FINITE-VOLUME ELEMENT METHOD FOR GENERAL SELF-ADJOINT ELLIPTICPROBLEMS

The finite volume element method (FVE) is a discretization technique f or partial differential equations. This paper develops discretization energy error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations, on which there exist linea r finite element spaces, and a very general type of control volumes (c ovolumes). The energy error estimates of this paper are also optimal b ut the restriction conditions for the covolumes given in [R. E. Bank a nd D. J. Rose, SIAM J. Numer. Anal., 24 (1987), pp. 777-787], [Z. Q. C ai, Numer. Math., 58 (1991), pp. 713-735] are removed. The authors fin ally provide a counterexample to show that an expected L-2-error estim ate does not exist in the usual sense. It is conjectured that the opti mal order of parallel to u - u(h) parallel to(0,Omega) should be O(h) for the general case.