A Neumann--Neumann Domain Decomposition Algorithm for Solving Plate and Shell Problems

We present a new Neumann–Neumann-type preconditioner of large scale linear systems arising from plate and shell problems. The advantage of the new method is a smaller coarse space than those of earlier methods of the authors; this improves parallel scalability. A new abstract framework for Neumann–Neumann preconditioners is used to prove almost optimal convergence properties of the method. The convergence estimates are independent of the number of subdomains, coefficient jumps between subdomains, and depend only polylogarithmically on the number of elements per subdomain. We formulate and prove an approximate parametric variational principle for Reissner–Mindlin elements as the plate thickness approaches zero, which makes the results applicable to a large class of nonlocking elements in everyday engineering use. The theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice.