# Optimal Combinations for Various Fems

• Zi Cai Li
Part of the Mathematics and Its Applications book series (MAIA, volume 444)

## Abstract

Let us study again the questions arising in Chapter 5. Suppose that the true solution is smooth such that uH k +1(S) where k ≥ 1, and S is the solution domain. Then the k-order Lagrange FEM can be used, and the resulting solutions and their generalized derivatives have the optimal convergence rate O(H k ), where H is the maximal length of the k-order quasiuniform elements. Quite often in practical problems, the smoothness of the true solution u is not uniform throughout the entire solution domain S. For instance, the solution u in one part, i.e., S2, of S is smoother than u in the rest, S1, of S:
$$u \in {H^{m + 1}}\left( S \right),u \in {H^{k + 1}}\left( {{S_2}} \right),1 \leqslant m < k,$$
where S = S1S2, S1S2 = ∅, and H m (S) is the Sobolev space. As a consequence, we can use the lower m-order Lagrange FEM in the entire solution domain S. By means of the assumption uH k +1(S2), we may also use the higher k-order Lagrange FEM in the subdomain S2 to gain a reduction of CPU time and computer storage, in particular when S2 is much larger than S1. Therefore, the use of FEM combinations is important in both theoretical and practical aspects. Unfortunately, the reduced rates of convergence are often caused by the nonconforming constraints in Chapter 5. Naturally a question arises: Can we find other coupling techniques in Chapters 7–9 to match different FEMs such that the optimal convergence rates can always be achieved? This chapter attempts to give a positive answer to this question.

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