Long Title Your Name Joint work with X (from here), Y (from here) - PowerPoint PPT Presentation

Long Title Your Name Joint work with X (from here), Y (from here) and Z (from here)

Outline 1 Section 1 2 Section 2 3 Section 3 4 Section 4 Your Name (Durham) Short Title 29 July 2010 2 / 14

Section 1 Your Name (Durham) Short Title 29 July 2010 3 / 14

Frame Title in Ω ⊂ R d − ε ∆ u + b · ∇ u = f u = 0 on ∂ Ω Look at these equations... Theorem A Theorem − 1 2 ∇ · b ≥ ρ ≥ 0 . Proof. A Proof � Your Name (Durham) Short Title 29 July 2010 4 / 14

Frame Title There are some pictures below... (a) ε = 0 . 1 (b) ε = 0 . 01 (c) ε = 0 . 001 Your Name (Durham) Short Title 29 July 2010 5 / 14

Another Frame Some equations and pictures... − 0 . 01∆ u + ( − 1 , 0) ⊤ · ∇ u = 1 in Ω ⊂ R d u = 0 on ∂ Ω 1 1 0.9 0.8 0.8 1 0.7 0.9 0.8 0.6 0.6 0.7 0.5 0.6 0.5 0.4 0.4 0.4 0.3 0.3 0.2 1 0.2 0.2 0.1 0.8 0 0.6 0.1 0 0.2 0.4 0.4 0 0 0.6 0.2 0 0.2 0.4 0.6 0.8 1 0.8 1 0 (a) Standard plot (b) Temperature map Your Name (Durham) Short Title 29 July 2010 6 / 14

Section 2 Your Name (Durham) Short Title 29 July 2010 7 / 14

Frame Title A definition... Definition (Something to Define) Some text... B ε ( u h , v ) := ε ( ∇ u h , ∇ v ) + ( b. ∇ u h , v ) = ( f, v ) ∀ v ∈ V h . Your Name (Durham) Short Title 29 July 2010 8 / 14

more? Your Name (Durham) Short Title 29 July 2010 9 / 14

Section 3 Your Name (Durham) Short Title 29 July 2010 10 / 14

[ · ℄ and average { [ ν ℄ = ν + n + + ν − n − , [ τ ℄ = τ + · n + + τ − · n − listing things... Γ the union of boundary faces (those in ∂ Ω ). h , T + is the downwind cell, T − the upwind cell as For e ∈ E o determined by b · n on the face from each cell, n being the outward pointing normal. [ ν ℄ = νn, For e ∈ E o h the jump {·} } are defined by [ ν ℄ · { [ τ ℄ } = 1 } = 1 2( ν + + ν − ) , 2( τ + + τ − ) . { { ν } { { τ } On the boundary these become { { ν } } = ν, { { τ } } = τ. A very useful identity � � � � ν τ · n = { τ } } + { { ν } } e ∈E o ∂T e ∈E h T ∈T h h Your Name (Durham) Short Title 29 July 2010 11 / 14

Section 4 Your Name (Durham) Short Title 29 July 2010 12 / 14

Future work on... Idea 1. Idea 2. Idea 3. Extra info... Your Name (Durham) Short Title 29 July 2010 13 / 14

References Your Name (Durham) Short Title 29 July 2010 14 / 14