Multilevel finite element preconditioning for $\sqrt{3}$ refinement

We develop a BPX-type multilevel method for the numerical solution of second order elliptic equations in $\mathbb{R}^2$ using piecewise linear polynomials on a sequence of triangulations given by regular $\sqrt{3}$ refinement. A multilevel splitting of the finest grid space is obtained from the nonnested sequence of spaces on the coarser triangulations using prolongation operators based on simple averaging procedures. The main result is that the condition number of the corresponding BPX preconditioned linear system is uniformly bounded independent of the size of the problem. The motivation to consider $\sqrt{3}$ refinement stems from the fact that it is a slower topological refinement than the usual dyadic refinement, and that it alternates the orientation of the refined triangles. Therefore we expect a reduction of the amount of work when compared to the classical BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement.