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

Error Bound Optimal Rate Triangular Element Finite Difference Method Admissible Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Kluwer Academic Publishers 1998