bifurcation problems where there is an interaction between two different modes. We indicate a general methodology Qualitatively incorrect bifurcation diagrams are computed unless the symmetries both of the solutions and of the underlying differen-for grid design using the symmetries of the underlying tial equation are properly accounted for. In the context of the compupartial differential equation and of the solutions it is detation of solutions to partial differential equations using the finite sired to compute.
element method, this requires careful thought when designing a
We assume that the particular solutions of interest are suitable computational domain and corresponding grid. Here we periodic: analysis (e.g. linear stability analysis) of the parconsider the problem of computing the interaction of hexagon and roll solutions bifurcating from a spatially uniform equilibrium solu-tial differential equation often indicates the wavenumbers tion of an E(2) equivariant partial differential equation. As an examof the solutions on the branches passing through the bifurple of where such an interaction occurs we consider a partial differcation point. A finite domain is chosen which will allow a ential equation describing the directional solidification of a dilute small number of wavelengths of these solutions; the dobinary alloy. We show that if the symmetry is not taken into account main can subsequently be extended by translation and then spurious disconnections can occur. We describe how to overcome this problem by constructing novel grids which have hexago-reflection to cover the whole plane. We note that if this is nal symmetry and enough translational symmetry to enable the to be done, then only geometries which tile the whole plane computation of the correct bifurcation structure. แฎ 1997 Academic Press can be used; for example, triangles, squares, rectangles, and hexagons.
The original partial differential equation on the infinite domain often has some symmetry: in the case we consider 18