Finite Element Methods for Linear Elasticity
نویسنده
چکیده
Conditions for Stable Approximation Schemes Basic idea: Mimic structure of continuous problem. To establish stability of continuous problem, only used last two spaces in top sequence and last three spaces in bottom sequence. Λn−1(K) dn−1 −−−→ Λn(K)→ 0 ↗ Sn−2 ↗ Sn−1 Λn−2(V) dn−2 −−−→ Λn−1(V) dn−1 −−−→ Λn(V)→ 0. Thus, look for five finite dimensional spaces connected by a similar structure, i.e., in addition to spaces Λh(K) ⊂ HΛ n(K), Λn−1 h (V) ⊂ HΛ n−1(V), Λh(V) ⊂ HΛ n(V) used in finite element method, also seek spaces Λn−1 h (K) ⊂ HΛ n−1(K), Λn−2 h (V) ⊂ HΛ n−2(V). 59 Require finite element spaces also connected by exact sequences, but introduce additional flexibility by inserting L2 projection operator Πh and using approximations of Sn−2 and Sn−1. Λn−1 h (K) Πhdn−1 −−−−−→ Λh(K)→ 0 ↗ Sn−2,h ↗ Sn−1,h (7) Λn−2 h (V) dn−2 −−−→ Λn−1 h (V) dn−1 −−−→ Λh(V)→ 0. Next step: Identify properties of interpolants into each finite element space needed for stability proof. Define Πh and Π̃ n h to be L 2 projection operators into Λh(K) and Λh(V), respectively.
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