Notes for “A Mixed Finite Element Method for 2-nd Order Elliptic Problems” by Raviart

and Thomas, 1977

James M. Rath

January 24, 2002

Abstract

Published in “Mathematical aspects of finite element methods : proceedings of the conference held in Rome, 10-12 December 1975” by Springer-Verlag in 1977. Part of the “Lecture Notes in Mathematics” series; it’s number 606.

Contents

1 Typographical Errors 3 2 Annotations 5

1 Typographical Errors

on page 293:

• line 3: “ptoblem” should read “problem” on page 295:

• line 2 of remark 1: “seddle-point” should read “saddle-point” on page 297:

• line 8: “consrtuction” should read “constructing”

• in equation 2.16: “K′ = K_{1} ⊂ K_{2}” should read “K′ = K_{1} ∩ K_{2}” on page 298:

• in lemma 1: the function is uniquely determined by both conditions a

and b on page 299:

• at the bottom, in remark 3: “… one could have equivalently specified the moments of order ≤ k” should read “… one could have equivalently specified the moments of order ≤ k − 1 and … ”

on page 300:

• in the top half of equation 3.8: “α_{1}ξ^{k}” should read “α_{1}ξ^{k}η”

• in the bottom half of equation 3.8: the open parenthesis next to the

β_{1}ξη^{k} term is spurious on page 302:

• in equation 3.21: “∥B_{K}∥^{2}” should read “∥B_{K}∥^{l}” on page 303:

• in theorem 3: “Then there exist an operator π_{∼}_{K} ∈ L((H^{1}(K))^{2}” should read “Then there exists an operator π_{∼}_{K} ∈ L((H^{1}(K))^{2}) such that π_{∼}_{K} :

(H^{1}(K))^{2} → Q_{∼}_{K}”

• equation 3.23 in theorem 3 should add the relation: ∀r_{∼} ∈ (P_{k−1})^{2} , ∫_{K} (π_{∼}_{K} q_{∼} − q_{∼})·

r_{∼} dx = 0

• in equation 3.26: the integral should be equal to 0

• in the proof of theorem 3: “… there exists by Lemma 1 and Remark 1

… ” should read “… there exists by Lemma 1 and Remark 3 … ” on page 305:

• in the second line after equation 4.2: “Note that the function … ”

should read “Note that the functions … ” on page 306:

• in equation 4.7: on the RHS of the inequality “u” should be “μ” on page 308:

• equation 4.19 is missing a closing brace at the end on page 311:

• in equation 4.38: “∥p_{∼} − p_{h}∥_{H_{∼}(div;Ω)}” should read “∥p_{∼} − p_{∼}_{h}∥_{H_{∼}(div;Ω)}”

• third line from the bottom: “π_{∼}_{h} ∈ Q_{∼}_{h}” should read “π_{∼}_{h}p ∈ Q_{∼}_{h}” on page 312:

• in the statement following equation 4.30: “… inequalities (2.11) … ” should read “… inequalities (2.13) … ”

2 Annotations

on page 294:

• equation 2.4: recall that p = grad u or p = ∇u

• equation 2.5: recall that − div p = f or −∇ · p = f on page 295:

• regarding the top two lines: that is, the divergence operator maps

H_{∼}(div, Ω) onto L^{2}(Ω) on page 296:

• equation 2.11 ⇒ H2

• equation 2.12 ⇔ H1 on page 298:

• equation 3.1 is an approximation condition

• equation 3.2 is an unisolvance condition

• equation 3.3 is equivalent to equation 2.11

• equation 3.4 is a conforming condition

• equation 3.5 is an unisolvance condition on page 300:

• the top half of equation 3.8 is equal to p_{k} + ∑_{j=0}^{k/2} α_{j}ξ^{k+1−j}η^{j}

• the bottom half of equation 3.8 is equal to p_{k} + ∑_{j=0}^{k/2} β_{j}η^{k+1−j}ξ^{j} on page 301:

• the top half of equation 3.11 is equal to p_{k} + ∑j=0^{\frac{k+1}{2}} α_{j}ξ^{k+1−j}η^{j}

• the bottom half of equation 3.11 is equal to p_{k} + ∑j=0^{\frac{k+1}{2}} β_{j}η^{k+1−j}ξ^{j} on page 303:

• The relations defined in equation 3.23 (as amended above) are the defining relations of the FEM used.

• in equation 3.28: ∥div (π̂_{∼}q̂_{∼} − q̂_{∼})∥_{0,K̂} = ∥P̂ (div q̂_{∼}) − (div q̂_{∼})∥_{0,K̂}

• regarding the spaces defined on this page, is the following commutative

(!) diagram: (H^{1}(K̂))^{2} ⟶^{div} L^{2} π̂_{∼}↓  _{P̂=}↓^{L^{2}−proj} Q̂_{∼}  ⟶^{div} P̂_{K} ⟶ 0

on page 304: • As a corrolary to theorem 3 on p303, we have the following commu-

tative (!) diagram: H^{1}(K) ⟶^{div} L^{2}(K) ⟶ 0 π_{∼}↓  ^{P=}↓^{L^{2}−proj} Q_{∼}  ⟶^{div} P_{k}(K) ⟶_{see4.3} 0 ↓  ↓ 0  0 and that ∇_{∼} · π_{∼}q_{∼} =

P (∇_{∼} · q_{∼}).

on page 309:

• regarding equations 4.23, 4.24, and the following choice of the charac-

teristic function: this is a hybrid method for the PDE

\{ \begin{matrix}−Δϕ & = & v & in Ω \\ ϕ & = & 0 & on ∂Ω \end{matrix}

on page 310:

• in the proof at the bottom: recall that Q_{∼}_{h} ⟶_{(onto)}^{div} V^{h} ⟶ 0 (div is bounded below for an appropriate inverse)