Mimetic Finite Difference Methods

Mimetic Finite Difference Methods Applications to the Diffusion Equation Asbjørn Nilsen Riseth May 15, 2014

2014-05-15 Mimetic Finite Difference Methods Mimetic Finite Difference Methods Applications to the Diffusion Equation Asbjørn Nilsen Riseth May 15, 2014 Done: Looked into MFD, scheme to solve PDEs. Implemented software in DUNe. Tested convergence results

Motivation Richards equation, pressure head and water content in soil θ(ϕ) t Diffusion equation div[k(ϕ) (ϕ + z)] = b div(k p) = b ``New'' numerical discretisation method Discrete vector calculus identities Convergence results for wide range of meshes 2

Diffusion equation, mixed form Ω R n Lipschitz. Find u C 1 (Ω; R n ), p C 1 (Ω) st u(x) = K p(x) x Ω div(u)(x) = b(x) x Ω p(x) = g D (x) x Γ D (u ˆn)(x) = g N (x) x Γ N 3

Diffusion equation, mixed form Ω R n Lipschitz. Find u H div (Ω), p H 1 (Ω) st K 1 u v dx p v dx Ω Ω + p v ˆn dσ = g D v ˆn dσ v H div (Ω) Γ N Γ D Unique solution if Γ D q, u L 2 (Ω) = q, b L 2 (Ω) q H0(Ω) 1 q, u ˆn L 2 (Γ N ) = q, g N L 2 (Γ N ) q L 2 (Γ N ) 4

Discrete Vector Calculus div(v) dx = Ω Ω v ˆn dσ Ω q div(v) dx = q v dx Ω Polygonal mesh - discrete spaces X and V p c := p dx v c, f := v ˆn c, f dx c Divergence operation DIV : V X (DIVv) c := 1 f v c, f c f F (c) (DIVv h ) c dx = f v c, f = c f F(c) f c v ˆn dx 5

2014-05-15 Mimetic Finite Difference Methods Discrete Vector Calculus Discrete Vector Calculus div(v) dx = v ˆn dσ q div(v) dx = q v dx Ω Ω Ω Ω Polygonal mesh - discrete spaces X and V pc := p dx vc, f := v ˆnc, f dx c f Divergence operation DIV : V X (DIVv)c := 1 f vc, f c f F (c) (DIVvh)c dx = f vc, f = v ˆn dx c f F(c) c Divergence theorem Zero boundary conditions, gradient and divergence adjoints If v h interpolated, divergence thm true

Discrete Vector Calculus Cell-wise inner products [p, q] X,c = c p c q c [v, u] V,c = v T c M V,c u c M V,c must satisfy consistency and stability [u c, v c ] V,c = Kc 1 u v dx certain u, v interpolates c v c v c [v c, v c ] V,c c v T c v c Define G : X V cell-wise c G c := M 1 V,c (DIV c) T c [q, DIVv] X = [Gq, v] V v c V c 6

2014-05-15 Mimetic Finite Difference Methods Discrete Vector Calculus Discrete Vector Calculus Cell-wise inner products [p, q]x,c = c pcqc [v, u]v,c = v T c MV,cuc MV,c must satisfy consistency and stability [uc, vc]v,c = c vcvc [vc, vc]v,c c v T c vc Define G : X V cell-wise c K 1 c u v dx certain u, v interpolates Gc := M 1 V,c (DIVc)T c [q, DIVv]X = [Gq, v]v vc Vc Consistentcy: u, div(u) constant on c, v n f face-wise constant Requirements give rise family of schemes. Depending on grid and matrix, same as finite differences and 0-order Raviart Thomas finite elements Gradient def: matrix representations of maps Don't actually invert M Summarise discrete vector calculus set-up

System of equations Deal with boundary conditions Face-values for pressure, λ f := p(x) dx f Flux-continuity v c, f = v c, f if f = c c [u c, v c ] V,c = c p c DIV c v c f λ f v c, f [DIVu c, q] X,c = q c b c c f F (c) 7

System of equations Deal with boundary conditions Face-values for pressure, λ f := p(x) dx f Flux-continuity v c, f = v c, f if f = c c M V,c u h,c = B T c p c C T c λ c B c u h,c = c b c 7

System of equations Deal with boundary conditions Face-values for pressure, λ f := p(x) dx f Flux-continuity v c, f = v c, f if f = c c M V B T C T v C T B 0 0 p = D gd M X b C 0 0 λ g N 7

MV B T CT 2014-05-15 Mimetic Finite Difference Methods System of equations System of equations Deal with boundary conditions Face-values for pressure, λ f := p(x) dx f Flux-continuity vc, f = vc, f if f = c c B 0 0 C 0 0 v p = λ C T D gd MXb g N For each vector v c and q of right size functions Gives matrix equation Take into account boundary conditions C T maps face-elements to local cell-wisepositions If Γ D, then this is invertible Schur factorisation twice gives a positive-definite matrix system

Post-processing and convergence Reconstruct cell-wise constant vector field u Set p (x) = p c Kc 1 u (x x c ) for x c Second-order convergence p p L 2 (Ω) Ch 2 ( p H 2 (Ω) + b H 1 (Ω)) 8

2014-05-15 Mimetic Finite Difference Methods Post-processing and convergence Post-processing and convergence Reconstruct cell-wise constant vector field u Set p (x) = pc Kc 1 u (x xc) for x c Second-order convergence p p L 2 (Ω) Ch 2 ( p H 2 (Ω) + b H 1 (Ω)) u c = 1 c RT c u c

``Test5'', Sine problem. Varying K-tensor p = sin(2πx) sin(2πy) + x 2 + y 2 + 1 ( (x + 1) K = 2 + y 2 ) xy xy (x + 1) 2 Γ N = cells h L 2 error L 2 eoc H 1 error H 1 eoc 256 0.125 0.0311225 1.9757 1.17476 0.96445 1024 0.0625 0.00781411 1.9938 0.590986 0.991173 4096 0.03125 0.00195564 1.99844 0.295946 0.997792 16384 0.015625 0.000489044 1.99961 0.14803 0.999447 65536 0.0078125 0.000122269 1.9999 0.0740219 0.999862 262144 0.00390625 3.05679e-05 1.99998 0.0370118 0.999965 Table: Convergence table for Test5. 9

2014-05-15 Mimetic Finite Difference Methods ``Test5'', Sine problem. Varying K-tensor ``Test5'', Sine problem. Varying K-tensor p = sin(2πx) sin(2πy) + x 2 + y 2 + 1 ( (x + 1) K = 2 + y 2 xy ) xy (x + 1) 2 Γ N = cells h L 2 error L 2 eoc H 1 error H 1 eoc 256 0.125 0.0311225 1.9757 1.17476 0.96445 1024 0.0625 0.00781411 1.9938 0.590986 0.991173 4096 0.03125 0.00195564 1.99844 0.295946 0.997792 16384 0.015625 0.000489044 1.99961 0.14803 0.999447 65536 0.0078125 0.000122269 1.9999 0.0740219 0.999862 262144 0.00390625 3.05679e-05 1.99998 0.0370118 0.999965 Table: Convergence table for Test5. Domain is just (0, 1) 2

A C 1 problem p(x) = 1 x 2 <0.5(cos(2π x 2 ) + 1) cells h L 2 error L 2 eoc H 1 error H 1 eoc 256 0.125 0.038842 0.619657 0.586107 0.84393 1024 0.0625 0.0145409 1.4175 0.293317 0.998702 4096 0.03125 0.00145493 3.32109 0.143665 1.02975 16384 0.015625 0.000685509 1.08571 0.07201 0.99644 65536 0.0078125 0.000166328 2.04315 0.0360368 0.998725 262144 0.00390625 0.000136436 0.285809 0.0180391 0.998341 Table: Convergence table for C 1 problem. 10

A C 1 problem p(x) = 1 x 2 <0.5(cos(2π x 2 ) + 1) cells h L 2 error L 2 eoc H 1 error H 1 eoc 256 0.125 0.038842 0.619657 0.586107 0.84393 1024 0.0625 0.0145409 1.4175 0.293317 0.998702 4096 0.03125 0.00145493 3.32109 0.143665 1.02975 16384 0.015625 0.000685509 1.08571 0.07201 0.99644 65536 0.0078125 0.000166328 2.04315 0.0360368 0.998725 262144 0.00390625 0.000136436 0.285809 0.0180391 0.998341 Table: Convergence table for C 1 problem. 10

Finite element comparisons 0.1 0.01 L 2 -error 0.001 0.0001 1e-05 0.01 0.1 h Test5 mimetic Test5 FE C1 mimetic C1 FE Figure: Approximation comparisons for ``Test5'' and C 1 problem. 11

Finite element comparisons 3 2.5 2 L 2 -eoc 1.5 1 0.5 0 Test5 mimetic Test5 FE C1 mimetic C1 FE 0.01 0.1 h Figure: Approximation comparisons for ``Test5'' and C 1 problem. 11

Outlook Richards equation More difficult meshes Generalise to other elliptic problems Performance comparisons 12

Outlook Richards equation More difficult meshes Generalise to other elliptic problems Performance comparisons Thank you 12