Abstract
Usually, the smoothness of solutions in the interior domain is higher, and the smoothness near the boundary is lower. One natural combination is to choose high order Lagrange-FEM in the interior subdomain, and lower order Lagrange-FEM in the boundary layer. This chapter displays the significance of combinations to solve not only the singularity problems but also the general elliptic equations. Also, such a combination of various FEMs is also a representative for many other kinds of combined methods using different admissible functions. The k— order Lagrange FEM can be used on the solution domain S if the solution u ∈ Hk +1(S). However we have to use the linear finite element method on S if u ∈ H2(S) only. The smoothness of the solutions of elliptic equations is, generally, different on different subdomains. For example, the solution u satisfies (Figure 5.1):
where S consists of two subdomains, S1 and S2. Since u ∈ H2(S) only, the linear FEM has to be used on S1 with relatively small linear elements. Clearly, the k-order Lagrange FEM ought to be used on S2 because u ∈ Hk +1(S2) (k ≥ 2). We hope that relatively large, k-order elements are applied in S2 such that calculation work and storage space can be saved.
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© 1998 Kluwer Academic Publishers
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Li, Z.C. (1998). Combinations of Various Fems. In: Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities. Mathematics and Its Applications, vol 444. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3338-8_5
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DOI: https://doi.org/10.1007/978-1-4613-3338-8_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3340-1
Online ISBN: 978-1-4613-3338-8
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