Uniform Substructuring Preconditioners for High Order FEM on Triangles and the Influence of Nodal Basis Functions

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1465-1491, August 2024.
Abstract. A robust substructuring type preconditioner is developed for high order approximation of problem for which the element matrix takes the form [math] where [math] and [math] are the mass and stiffness matrices, respectively. A standard preconditioner for the pure stiffness matrix results in a condition number bounded by [math] where [math] blows up as [math]. It is shown that the best uniform bound in [math] that one can hope for is [math]. More precisely, we show that the upper envelope of the bound [math] is [math]. What, then, can be done to obtain a preconditioner that is robust for all [math]? The solution turns out to be a relatively minor modification of the basic substructuring algorithm of [I. Babuška et al., SIAM J. Numer. Anal., 28 (1991), pp. 624–661]: one can simply augment the preconditioner with a suitable Jacobi smoothener over the coarse grid degrees of freedom. This is shown to result in a condition number bounded by [math] where the constant is independent of [math]. Numerical results are given which shows that the simple expedient of augmentation with nodal smoothening reduces the condition number by a factor of up to two orders of magnitude.