Jaeryun YIM from Seoul National University My works: 1 P 1 - - PowerPoint PPT Presentation

Jaeryun YIM from Seoul National University My works:

1 P1 Nonconforming FE

• piecewise linear function on quadrilateral element
• continuous at midpoint of each edge

2 Residual Based A Posteriori Error Estimator for P1 NCFE

η2 :=

• K

h2

K f + div∇unc h

• cell residual

2 0,K +

• E

hE

• JE,ν2

0,E

• normal jump

+ JE,τ2

0,E

• tangential jump
• (1)

where JE,ν :=      ν1 · ∇unc

h |K1 + ν2 · ∇unc h |K2

E ∈ E(Ω) g − ν · ∇unc

h

E ∈ E(ΓN) E ∈ E(ΓD) (2) JE,τ :=      τ1 · ∇unc

h |K1 + τ2 · ∇unc h |K2

E ∈ E(Ω) E ∈ E(ΓN) τ · (∇uD − ∇unc

h )

E ∈ E(ΓD) (3)

1 / 3

Consider −△u = 0 in Ω = [−1, 1]2\[0, 1]2 Exact solution in polar coordinate p(r, θ) = rα sin(α(θ − π/2)) ∈ H1+α(Ω), α = 2/3 (4)

• 0.6
• 0.4
• 0.2

• 0.6
• 0.4
• 0.2

• 0.1

• 0.6
• 0.4
• 0.2

• 0.6
• 0.4
• 0.2

• 0.1

Figure : (Tops) P1 NCFE Solution w/ tangential term: (L) After 1 refinement, (R) After 7

(Bottoms) Grid after 7 refinements: (L) Q1 CFE, (C) P1 NCFE w/o tangential term, (R) P1 NCFE w/ tangential term

2 / 3

Further works: tangential jump term on Dirichlet boundary edges coefficient consideration for the general elliptic problem a posterior error estimator for NCFE to another problems

3 / 3