Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities pp 309-326 | Cite as

# Optimal Combinations for Various Fems

Chapter

## Abstract

Let us study again the questions arising in Chapter 5. Suppose that the true solution is smooth such that
where

*u*∈*H*^{ 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.,*S*_{2}, of*S*is smoother than*u*in the rest,*S*_{1}, of*S*:$$
u \in {H^{m + 1}}\left( S \right),u \in {H^{k + 1}}\left( {{S_2}} \right),1 \leqslant m < k,
$$

*S*=*S*_{1}∪*S*_{2},*S*_{1}∩*S*_{2}= ∅, 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*u*∈*H*^{ k }^{+1}(*S*_{2}), we may also use the higher*k*-order Lagrange FEM in the subdomain*S*_{2}to gain a reduction of CPU time and computer storage, in particular when*S*_{2}is much larger than*S*_{1}. 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.### Keywords

Cond## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Kluwer Academic Publishers 1998