GRID DESIGN FOR THE COMPUTATION OF A HEXAGON-ROLL INTERACTION USING AFINITE-ELEMENT METHOD

Qualitatively incorrect bifurcation diagrams are computed unless the s ymmetries both of the solutions and of the underlying differential equ ation are properly accounted for. In the context of the computation of solutions to partial differential equations using the finite element method, this requires careful thought when designing a suitable comput ational domain and corresponding grid. Here we consider the problem of computing the interaction of hexagon and roll solutions bifurcating f rom a spatially uniform equilibrium solution of an E(2) equivariant pa rtial differential equation. As an example of where such an interactio n occurs we consider a partial differential equation describing the di rectional solidification of a dilute binary alloy. We show that if the symmetry is not taken into account then spurious disconnections can o ccur. We describe how to overcome this problem by constructing novel g rids which have hexagonal symmetry and enough translational symmetry t o enable the computation of the correct bifurcation structure. (C) 199 7 Academic Press.