Combinations of Various Fems

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 ∈ H^{k +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):

$$ u \in {{H}^{2}}\left( S \right),u \notin {{H}^{3}}\left( S \right)andu \in {{H}^{{k + 1}}}\left( {{{S}_{2}}} \right)\left( {k \geq 2} \right), $$

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 ∈ H^{k +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.