High-order finite volume method for curved boundaries and non-matching domain-mesh problems May 23-27, 2016, S˜ ao F´ elix, Portugal
Ricardo Costa 1,2 , St´ephane Clain 1,2 , Gaspar J. Machado 2 , Rapha¨el Loub`ere 1 1 Institut 2 Centro
de Math´ ematiques de Toulouse, Universit´ e de Toulouse, France de Matem´ atica, Universidade do Minho, Guimar˜ aes, Portugal
Outline
1 Motivation and Background 2 Problem Formulation 3 Polynomial Reconstruction Machinery 4 ARCH Method 5 Finite Volume Scheme 6 Numerical Benchmark 7 Conclusions and Final Remarks
2/28
Motivation and Background
Replacing curved boundaries by polygonal edges associated to the mesh provides at most 2nd-order accuracy
∂Ω O(h2) M
Isoparametric elements are widely applied for FEM and DG Very few methods have been developed in the context of HO-FVM
Motivation and Background
3/28
State-of-art in HO-FVM Ghost cells approach: add extra cells between the geometric boundary and the computational domain
1 C.F.
Ollivier-Gooch and M. Van Altena, A high-order accurate unstructured
mesh finite-volume scheme for the advection-diffusion equation, Journal of Computational Physics, (2002). Motivation and Background
4/28
State-of-art in HO-FVM Ghost cells approach: add extra cells between the geometric boundary and the computational domain Ollivier-Gooch approach1 : enforce the BC by constraining the LSM associated to the PR can be very time consuming if the LSM matrix has to be updated (moving boundaries/interfaces, tracking interfaces/discontinuities problems)
1 C.F.
Ollivier-Gooch and M. Van Altena, A high-order accurate unstructured
mesh finite-volume scheme for the advection-diffusion equation, Journal of Computational Physics, (2002). Motivation and Background
4/28
State-of-art in HO-FVM Ghost cells approach: add extra cells between the geometric boundary and the computational domain Ollivier-Gooch approach1 : enforce the BC by constraining the LSM associated to the PR can be very time consuming if the LSM matrix has to be updated (moving boundaries/interfaces, tracking interfaces/discontinuities problems)
New approach: detach the BC conservation from the LSM matrix Easier, flexible, more efficient, more elegant 1 C.F.
Ollivier-Gooch and M. Van Altena, A high-order accurate unstructured
mesh finite-volume scheme for the advection-diffusion equation, Journal of Computational Physics, (2002). Motivation and Background
4/28
Problem Formulation Poisson’s equation and BCs: ∇2 φ = f ,
in Ω
φ = φD ,
on ΓD
∇φ · n = gN ,
on ΓN
αφ + β∇φ · n = gR ,
on ΓR
Ω, ∂Ω = {ΓD ∪ ΓN ∪ ΓR } – real domain and its boundary n – unit normal vector to ∂Ω φD – Dirichlet BC gN – Neumann BC gR – Robin BC with coefficients α and β Problem Formulation
5/28
Generic FV Scheme ΓD
ei D ni D
ci , eij – cell, edge eij
ci
nij
mi
qij,r
qi D,r
Z
mj cj
nij – normal vector "
Z (∇φ) · ni ds =
∂ci
X
f dx ⇒ ci
qi , qij –quadrature points
|eij |
j∈ν(i)
R X
# ζr Fij,r − fi |ci | = O(hi2R )
r =1
Physical fluxes: Fij,r = ∇φ(qij,r ) · nij , Source term: fi =
S X
ζs f (qi,s )
s=1 Problem Formulation
6/28
Polynomial Reconstruction Machinery Non-conservative PR for inner edges eij ϕij (x ) =
X
α Rα ij (x − mij )
0≤|α|≤d
α = (α1 , α2 ), |α| = α1 + α2 , x α = x1α1 x2α2 Rij = (Rα ij )0≤|α|≤d – coefficients (to be determined)
Polynomial Reconstruction Machinery
7/28
Polynomial Reconstruction Machinery Non-conservative PR for inner edges eij ϕij (x ) =
X
α Rα ij (x − mij )
0≤|α|≤d
α = (α1 , α2 ), |α| = α1 + α2 , x α = x1α1 x2α2 Rij = (Rα ij )0≤|α|≤d – coefficients (to be determined)
Eij (Rij ) =
X q∈Sij
ñ ωij,q
1 |cq |
ô2
Z ϕij (x ) dx − φq cq
‹ij = arg min [Eij (Rij )] eij (x ) defined with R ϕ Rij
Polynomial Reconstruction Machinery
7/28
Polynomial Reconstruction Machinery Conservative PR for Dirichlet boundary edges eiD ϕiD (x ) = φiD +
X
α α Rα iD [(x − miD ) − MiD ]
1≤|α|≤d
α = (α1 , α2 ), |α| = α1 + α2 , x α = x1α1 x2α2 RiD = (Rα iD )1≤|α|≤d – coefficients (to be determined)
Polynomial Reconstruction Machinery
8/28
Polynomial Reconstruction Machinery Conservative PR for Dirichlet boundary edges eiD ϕiD (x ) = φiD +
X
α α Rα iD [(x − miD ) − MiD ]
1≤|α|≤d
α = (α1 , α2 ), |α| = α1 + α2 , x α = x1α1 x2α2 RiD = (Rα iD )1≤|α|≤d – coefficients (to be determined)
EiD (RiD ) =
X q∈SiD
ñ ωiD,q
1 |cq |
ô2
Z ϕiD (x ) dx − φq cq
“iD = arg min [EiD (RiD )] biD (x ) defined with R ϕ RiD
Polynomial Reconstruction Machinery
8/28
Polynomial Reconstruction Machinery Naive mean-value conservation – 2nd-order! Z Z 1 1 φiD = φD (x ) ds, ϕ (x ) ds = φiD |eiD | eiD |eiD | eiD iD Z 1 α MiD = (x − miD )α dx |eiD | eiD
Polynomial Reconstruction Machinery
9/28
Polynomial Reconstruction Machinery Naive mean-value conservation – 2nd-order! Z Z 1 1 φiD = φD (x ) ds, ϕ (x ) ds = φiD |eiD | eiD |eiD | eiD iD Z 1 α MiD = (x − miD )α dx |eiD | eiD Wise mean-value conservation – Quadrature points on ∂Ω! Z Z 1 1 φiD = φD (x ) ds, ϕ (x ) ds = φiD |Û eiD | ÛeiD |Û eiD | ÛeiD iD Z 1 α MiD = (x − miD )α dx |Û eiD | ÛeiD
Polynomial Reconstruction Machinery
9/28
Polynomial Reconstruction Machinery Naive mean-value conservation – 2nd-order! Z Z 1 1 φiD = φD (x ) ds, ϕ (x ) ds = φiD |eiD | eiD |eiD | eiD iD Z 1 α MiD = (x − miD )α dx |eiD | eiD Wise mean-value conservation – Quadrature points on ∂Ω! Z Z 1 1 φiD = φD (x ) ds, ϕ (x ) ds = φiD |Û eiD | ÛeiD |Û eiD | ÛeiD iD Z 1 α MiD = (x − miD )α dx |Û eiD | ÛeiD Wise point-value conservation: φiD = φD (piD ), piD ∈ Û eiD ,
ϕiD (piD ) = φiD
α MiD = (piD − miD )α Polynomial Reconstruction Machinery
9/28
Polynomial Reconstruction Machinery Conservation of Dirichlet BC only α Coefficients MiD are boundary dependent
+ LSM matrix has to be updated for... ... moving boundaries/interfaces or dynamic BC in time-dependent and unsteady problems ... optimization problems ... tracking interfaces/discontinuities problems ... etc.
Ollivier-Gooch method consists in an augmented LSM matrix by constraints rows (equivalent to the wise conservation)
Polynomial Reconstruction Machinery
10/28
ARCH Method
ARCH Adaptive Reconstruction for Conservation of High-order
The aim of ARCH is to improve the boundary treatment The main ideas are... ... conserve the HO accuracy ... detach the boundary from the LSM matrix ... easy handling of moving boundaries ... generic treatment of Dirichlet, Neumann and Robin BCs ARCH Method
11/28
ARCH Method ARCH for boundary edges eiB ϕiB (x ; ψiB ) = ψiB + ϕiB (x ) ϕiB (x ) – non-conservative/naive conservative/wise conservative PR ψiB – free parameter (to be determined)
EiB (RiB ; ψiB ) =
X q∈SiB
ñ ωiB,q
1 ψiB + |cq |
ô2
Z ϕiB (x ; ψiB ) dx − φq cq
“ b ϕiB (x ; ψiB ) defined with RiB = arg min [EiB (RiB ; ψiB )] RiB
ψiB is a RHS – the LSM matrix remains the same :D ARCH Method
12/28
ARCH Method Generic BC on ∂Ω to satisfy: g(x ; α, β) = α(x )φ(x ) + β(x )∇φ · n Dirichlet BC: α = 1, β = 0 Neumann BC: α = 0, β 6= 0 Robin BC: α 6= 0, β 6= 0
ARCH Method
13/28
ARCH Method Generic BC on ∂Ω to satisfy: g(x ; α, β) = α(x )φ(x ) + β(x )∇φ · n Dirichlet BC: α = 1, β = 0 Neumann BC: α = 0, β 6= 0 Robin BC: α 6= 0, β 6= 0
For a given piB ∈ ∂Ω, the parameter ψiB is prescribed such that b b α(piB )ϕiB (piB ; ψ iB ) + β(piB )∇ϕiB (piB ; ψ iB ) · n = g(piB ; α, β) b 0 ψ iB = ψiB −
0 0 1 B(piB ; α, β, ψiB )(ψiB − ψiB ) 0 0 + ϕ1 ) B(piB ; α, β, ψiB ) − B(piB ; α, β, ψiB iB
0 1 ψiB 6= ψiB ARCH Method
13/28
Finite Volume Scheme Numerical fluxes: inner edge eij : Fij,r = ∇ϕ eij (qij,r b ) · nij
boundary edge eiB : FiB,r = ∇ϕiB (qiD,r ; ψiB ) · niB
PR and fluxes are linear identities Linear residual operator Φ → Gi (Φ) for vector Φ ∈ RI Gi (Φ) =
X j∈ν(i)
|eij |
" R X
# ζr Fij,r − fi |ci |,
r =1
G(Φ) = (Gi (Φ))i=1,...,I Free matrix method Compute vector Φ? ∈ RI solution of G(Φ) = 0 Finite Volume Scheme
14/28
Numerical Benchmark
Numerical Benchmark
15/28
Numerical Benchmark 1.2
Manufactured solution: φ(r ) = a + b ln(r )
0.6
x2
a, b such that φ ∈ [0, 1]
0
-0.6
-1.2 -1.2
-0.6
0
0.6
1.2
x1
Naive PR 1E0
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-2
Error
1E-4 1E-6
P1 P1 P3 P3 P5 P5
1E-8 1E-10 1E-12
6 4 2 1
1E3
Numerical Benchmark
1E4 DOF
1E5
16/28
Numerical Benchmark 1.2
Manufactured solution: φ(r ) = a + b ln(r )
0.6
x2
a, b such that φ ∈ [0, 1]
0
-0.6
-1.2 -1.2
-0.6
0
0.6
1.2
x1
Wise PR
1E-2
1E-2
1E-4
1E-4
1E-6
1E-12
1E-6
ARCH 1E0
P1 P1 P3 P3 P5 P5
6 4 2
1E-10 1
1E3
Numerical Benchmark
1E4 DOF
1E5
1E-12
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-2 1E-4
1E-8
1E-8 1E-10
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
Error
1E0
Error
Error
Naive PR 1E0
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 6 4 2
1E-10 1
1E3
1E4 DOF
1E5
1E-12
6 4 2 1
1E3
1E4 DOF
1E5
16/28
Numerical Benchmark Wise PR 1.2
ARCH
1E0
1E0
1E-2
1E-2
1E-4
1E-4
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
0
Error
x2
Error
0.6
1E-6 1E-8
-0.6
1E-10 -1.2 -1.2
-0.6
0 x1
Numerical Benchmark
0.6
1.2
1E-12
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 6 4 2
1E-10 1
1E3
1E4 DOF
1E5
1E-12
6 4 2 1
1E3
1E4 DOF
1E5
17/28
Numerical Benchmark Wise PR 1.2
ARCH
1E0
1E0
1E-2
1E-2
1E-4
1E-4
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
0
Error
x2
Error
0.6
1E-6 1E-8
-0.6
1E-10 -1.2 -1.2
-0.6
0
0.6
1E-10 1
1E3
1E4 DOF
x1
1E5
1E3
1E4 DOF
1E0
1E0
1E-2
1E-2
1E-4
1E-4
1E-6 1E-8
-0.6
1E-10 -1.2 -1.2
-0.6
0 x1
Numerical Benchmark
0.6
1.2
1E-12
P1 P1 P3 P3 P5 P5
1E5
ARCH
Error
Error
0
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1
1E-12
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
0.6
x2
P1 P1 P3 P3 P5 P5
6 4 2
Wise PR 1.2
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-8 6 4 2
1E-12
1.2
1E-6
P1 P1 P3 P3 P5 P5
1E-6
P1 P1 P3 P3 P5 P5
1E-8 6 4 2
1E-10 1
1E3
1E4 DOF
1E5
1E-12
6 4 2 1
1E3
1E4 DOF
1E5
17/28
Numerical Benchmark
DOF = 197340
Naive PR Wise PR ARCH
P1 E1 E∞ 2.91E−03 9.36E−03 1.20E−05 6.12E−05 1.20E−05 6.37E−05
P3 E1 E∞ 2.91E−03 9.36E−03 4.45E−09 3.60E−08 5.35E−09 3.74E−08
P5 E1 E∞ 2.91E−03 9.36E−03 2.98E−11 4.60E−10 9.48E−12 2.15E−09
DOF = 197340
Naive PR Wise PR ARCH
Numerical Benchmark
P1 E1 E∞ 2.87E−03 9.33E−03 1.07E−05 1.36E−02 1.03E−05 2.03E−03
P3 E1 E∞ 2.87E−03 9.31E−03 5.45E−09 1.16E−06 7.42E−10 4.40E−08
P5 E1 E∞ 2.87E−03 9.31E−03 3.37E−12 1.44E−10 1.93E−12 1.00E−10
18/28
Numerical Benchmark Wise PR 1.2
ARCH
1E0
1E0
1E-2
1E-2
1E-4
1E-4
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
0
Error
x2
Error
0.6
1E-6 1E-8
-0.6
1E-10 -1.2 -1.2
-0.6
0 x1
Numerical Benchmark
0.6
1.2
1E-12
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 6 4 2
1E-10 1
1E3
1E4 DOF
1E5
1E-12
6 4 2 1
1E3
1E4 DOF
1E5
19/28
Numerical Benchmark Wise PR 1.2
ARCH
1E0
1E0
1E-2
1E-2
1E-4
1E-4
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
0
Error
x2
Error
0.6
1E-6 1E-8
-0.6
1E-10 -1.2 -1.2
-0.6
0
0.6
1E-10 1
1E3
1E4 DOF
x1
1E5
1E3
1E4 DOF
1E0
1E0
1E-2
1E-2
1E-4
1E-4
1E-6 1E-8
-0.6
1E-10 -1.2 -1.2
-0.6
0 x1
Numerical Benchmark
0.6
1.2
1E-12
P1 P1 P3 P3 P5 P5
1E5
bfbfARCH
Error
Error
0
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1
1E-12
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
0.6
x2
P1 P1 P3 P3 P5 P5
6 4 2
Wise PR 1.2
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-8 6 4 2
1E-12
1.2
1E-6
P1 P1 P3 P3 P5 P5
1E-6
P1 P1 P3 P3 P5 P5
1E-8 6 4 2
1E-10 1
1E3
1E4 DOF
1E5
1E-12
6 4 2 1
1E3
1E4 DOF
1E5
19/28
Numerical Benchmark
DOF = 197340
Naive PR Wise PR ARCH
P1 E1 E∞ 5.88E−03 1.88E−02 6.47E−05 1.77E−04 6.48E−05 1.80E−04
P3 E1 E∞ 5.88E−03 1.88E−02 3.68E−08 1.37E−07 4.44E−08 1.76E−07
P5 E1 E∞ 5.88E−03 1.88E−02 1.80E−10 6.22E−09 1.21E−10 1.24E−08
DOF = 13056
Naive PR Wise PR ARCH
Numerical Benchmark
P1 E1 E∞ 2.13E−02 7.06E−02 9.72E−04 9.67E−03 9.24E−04 6.35E−03
P3 E1 E∞ 2.12E−02 7.05E−02 8.25E−06 6.20E−04 7.29E−07 8.27E−05
P5 E1 E∞ 2.12E−02 7.05E−02 3.31E−08 1.30E−06 4.84E−09 5.86E−07
20/28
Numerical Benchmark Wise PR
x2
Error
0.6
0
1E2
1E0
1E0
1E-2
1E-2
1E-4
1E-4
1E-6 1E-8
-0.6
1E-10 -1.2 -1.2
ARCH
1E2
Error
1.2
-0.6
0
0.6
1.2
1E-12
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 6 4 2
1E-10 1
1E3
x1
1E4 DOF
1E5
1E-12
6 4 2 1
1E3
1E4 DOF
1E5
DOF = 13056
Naive PR Wise PR ARCH
Numerical Benchmark
P1 E1 E∞ 2.13E−02 7.06E−02 9.72E−04 9.67E−03 9.24E−04 6.35E−03
P3 E1 E∞ 2.12E−02 7.05E−02 8.25E−06 6.20E−04 7.29E−07 8.27E−05
P5 E1 E∞ 2.12E−02 7.05E−02 3.31E−08 1.30E−06 4.84E−09 5.86E−07
21/28
Numerical Benchmark
ΓI , ΓE – interior and exterior boundaries x1 cos(θ) ΓI : = RI (θ; αI ) , x2 sin(θ) Å ã 1 RI (θ; αI ) = rI 1 + sin(αI θ) , 10
x1 cos(θ) ΓE : = RE (θ; αE ) x2 sin(θ) Å ã 1 RE (θ; αE ) = rE 1 + sin(αE θ) 10
Manufactured solution: φ(r , θ) = a(θ) + b(θ) ln(r ) a(θ) and b(θ) are chosen such that φ ∈ [0, 1] in Ω
Numerical Benchmark
22/28
Numerical Benchmark 1.2
0.6
x2
0
-0.6
-1.2 -1.2
-0.6
0
0.6
1.2
x1
Numerical Benchmark
23/28
Numerical Benchmark 1.2
0.6
x2
0
-0.6
-1.2 -1.2
-0.6
0
0.6
1.2
x1
Wise PR
1E-2
1E-2
1E-4
1E-4
1E-6 1E-8 1E-10 1E-12
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-6
ARCH 1E0
P1 P1 P3 P3 P5 P5
1E-4
1E-8 6 4 2
1E-10 1
1E3
Numerical Benchmark
1E4 DOF
1E5
1E-12
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-2
Error
1E0
Error
Error
Naive PR 1E0
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 6 4 2
1E-10 1
1E3
1E4 DOF
1E5
1E-12
6 4 2 1
1E3
1E4 DOF
1E5
23/28
Numerical Benchmark 1.2
0.6
x2
0
-0.6
-1.2 -1.2
-0.6
0
0.6
1.2
x1
Wise PR
1E-2
1E-2
1E-4
1E-4
1E-6 1E-8 1E-10 1E-12
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-6
ARCH 1E0
P1 P1 P3 P3 P5 P5
1E-4
1E-8 6 4 2
1E-10 1
1E3
Numerical Benchmark
1E4 DOF
1E5
1E-12
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
1E-2
Error
1E0
Error
Error
Naive PR 1E0
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 6 4 2
1E-10 1
1E3
1E4 DOF
1E5
1E-12
6 4 2 1
1E3
1E4 DOF
1E5
24/28
Numerical Benchmark | Mesh Pick-up
1.2
0.6
x2
0
-0.6
-1.2 -1.2
-0.6
0
0.6
1.2
x1
Numerical Benchmark
25/28
Numerical Benchmark | Mesh Pick-up Wise PR 1E0 1E-2
1.2
Error
1E-4
0.6
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 1E-10
x2
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
6 4 2 1
1E-12
1E3
0
1E4 DOF
1E5
ARCH 1E0 1E-2
-0.6
Error
1E-4
-1.2 -1.2
-0.6
0 x1
0.6
1.2
1E-6 1E-8 1E-10 1E-12
Numerical Benchmark
6 4 2 1
1E3
1E4 DOF
1E5
25/28
Numerical Benchmark | Mesh Pick-up Wise PR 1E0 1E-2
1.2
Error
1E-4
0.6
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 1E-10
x2
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
6 4 2 1
1E-12
1E3
0
1E4 DOF
1E5
ARCH 1E0 1E-2
-0.6
Error
1E-4
-1.2 -1.2
-0.6
0 x1
0.6
1.2
1E-6 1E-8 1E-10 1E-12
Numerical Benchmark
6 4 2 1
1E3
1E4 DOF
1E5
26/28
Numerical Benchmark | Mesh Pick-up Wise PR 1E0 1E-2
1.2
Error
1E-4
0.6
1E-6
P1 P1 P3 P3 P5 P5
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
P1 P1 P3 P3 P5 P5
1E-8 1E-10
x2
E1 , E∞ , E1 , E∞ , E1 , E∞ ,
6 4 2 1
1E-12
1E3
0
1E4 DOF
1E5
ARCH 1E0 1E-2
-0.6
Error
1E-4
-1.2 -1.2
-0.6
0 x1
0.6
1.2
1E-6 1E-8 1E-10 1E-12
Numerical Benchmark
6 4 2 1
1E3
1E4 DOF
1E5
27/28
Conclusions and Final Remarks
V HO is fully restored with the ARCH method V Wise PR and ARCH methods have comparable accuracy V Extrapolation situations are less robust
Conclusions and Final Remarks
28/28
Conclusions and Final Remarks
V HO is fully restored with the ARCH method V Wise PR and ARCH methods have comparable accuracy V Extrapolation situations are less robust I Benchmarks for Neumann and Robin BCs are in progress (promising results) I Improved mesh pick-up algorithms are in progress
Conclusions and Final Remarks
28/28
Conclusions and Final Remarks
V HO is fully restored with the ARCH method V Wise PR and ARCH methods have comparable accuracy V Extrapolation situations are less robust I Benchmarks for Neumann and Robin BCs are in progress (promising results) I Improved mesh pick-up algorithms are in progress
Obrigado Conclusions and Final Remarks
28/28
