An approach to improve Convergence Order in Finite Volume Methods and its Application in Finite Element Methods

We consider a second order elliptic problem posed on an interval or on a connected polygonal domain in a two dimensional space. We introduce an admissible mesh in the sense of [1]. The convergence order of the finite volume approximate solution (called Basic Finite Volume Solution) is in general $O(h)$ in both discrete norms $L^2$ and $H^1$. We suggest a technique, based on the so-called Fox's Difference Correction [2], which allows us to obtain a new Finite Volume Approximation of order $O(h^\alpha)$, where $\alpha$ equal to $2$ or $\frac{3}{2}$. In addition, this new Finite Volume Approximation can be computed using the same matrix used to compute the basic finite volume solution. The computational cost is comparable to that of the new Finite Volume Approximation. If the domain problem is an interval or a rectangle, we obtain finite volume approximations of arbitrary order and these approximations can be computed using the same matrix used to compute the Basic Finite Volume Solution. We give an application of our approach to improve the convergence order of finite element solutions defined in non uniform meshes.

References

1. R. Eymard, T. Gallouet and R. Herbin: Finite Volume Methods. Handbook of Numerical Analysis. P. G. Ciarlet and J. L. Lions (eds.), vol. VII, 723-1020, 2000.
2. L. Fox: Some Improvements in the Use of Relaxation Methods for the Solution of Ordinary and Partial Defferential Equations. Proc. Roy. Soc. Lon Ser. A, 190, 31-59, 1947.