Uniform substructuring preconditioners for high order FEM on triangles and the influence of nodal basis functions (Q6573778)

The authors introduce a new robust substructuring preconditioner for high-order FEM discretizations of problems in the form \(A_\kappa := (1 - \kappa)L + \kappa M\) where \(L\) is the stiffness matrix and \(M\) is the mass matrix.\N\NSection 3 investigates the influence of the choice of the basis functions on the preconditioner for the pure mass matrix case.\N\NFor arbitrary \(0 \le \kappa \le 1\), Section 4 presents a preconditioner that is robust with respect to \(\kappa\) and results in a bound of \(\mathcal{O}(1 + \log^2(p))\) for the condition number. This is achieved by combining an existing stiffness matrix substructuring preconditioner [\textit{I. Babuška} et al., SIAM J. Numer. Anal. 28, No. 3, 624--661 (1991; Zbl 0754.65083)] with an appropriate Jacobi smoothing step.\N\NNumerical experiments in Section 5 support the theoretical findings and are followed by the proofs of the main results in Sections 6--8. A summary in Section 9 concludes the paper.