sci.math.research SUNA13, Theorem & Corollaries ============================= This article is part of the "Series on Unified Numerical Approximations" UNIFICATION OF FINITE VOLUME AND GALERKIN METHODS FOR CONSERVATION EQUATIONS, IN THE TWO DIMENSIONAL CASE PART III : THE THEOREM In the previous article, an almost completed proof was given for the following First Unification Theorem ------------------------- | There exists an EQUIVALENCE between Finite Domain methods at Five Point Stars | and Galerkin methods at Quadrilaterals, when both applied to the conservation | equation in two dimensions, as formulated in general curvilinear coordinates. | | A necessary condition for the Galerkin method is, however, that the latter is | carried out with numerical integration at the _vertices_ of its Quadrilateral | elements (with is rather unusual). | | This unusual integration procedure is equivalent, in turn, with discretizing | at a set of mutually overlapping linear triangular elements. The integration | points of them must lie at the vertices of corresponding quadrilaterals. Suppose namely that we decide to evaluate the localized Galerkin integral at a linear triangle, which must be an even simpler procedure, then: dF/dh = F2 - F1 ; dF/dk = F3 - F1 ; and the integral equals: | - Qh - Qk | - | F1 F2 F3 | . | + Qh | | Qk | Here the Q's are evaluated at the (only) integration point of the triangle. Part of the corresponding formula in "SUNA, Theorem Proving" reads: h = 0 , k = 0 3 * ------- * 4 Triangles: ------------- | \ / | 1,2,3 | - Qh1 - Qk1 | | \ / | 2,4,1 - 1/4. [ | + Qh1 - 0 | + ... | / \ | 3,1,4 | - 0 + Qk1 | | / \ | 4,3,2 | + 0 + 0 | 1 * ------- * 2 The two terms are equal iff the integration point of the triangle attached at the quadrilateral node (1) precisely coincides with node (1); same argument for all of the four triangles. It is concluded therefrom that a quadrilateral element, with integration points at its corners, can entirely be replaced by a set of four overlapping triangles with their integration points at the vertices of the quadrilateral. The F.D.-F.E. Unification of the 2-D conservation equation problem as a _whole_ thus requires two, mutually _overlapping_, meshes of triangles, equipped with rather unusual integration points. This completes the proof of the First Unification Theorem. Some consequences: - The F.E. problem how to make the "best" choice between different triangular patterns in a rectangular grid is circumvented by the F.D. point of view: there exists *no* preferred pattern within a mesh of overlapping triangles. - Finite Differences are intrinsically more *in*efficient than Finite Elements, since they require a subdivision of quadrilaterals into *four* triangles, while F.E. normally does its job with only *two* triangles per quadrilateral. (This will become noticable only in the curvilinear case, however, since the way in which the computations are actually organized plays a dominant role.) - The fact that Finite Element meshes can be built up from overlapping elements suggests that even more unusual topologies *can* result in numerical schemes that still work, and there is a chance that they sometimes are even better. For pure Diffusion of heat, approximations of the fluxes (Qh,Qk) are constants. Then the position of the triangle integration points becomes _irrelevant_, and only the idea of overlapping elements survives. To be continued ... - * Han de Bruijn; Applications&Graphics | "A little bit of Physics * No * TUD Computing Centre; P.O. Box 354 | would be NO idleness in * Oil * 2600 AJ Delft; The Netherlands. | Mathematics" (HdB). * for * E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood