Additive and Multiplicative Schwarz Preconditioners for ... · Additive and Multiplicative Schwarz...

Additive and Multiplicative Schwarz Preconditioners forDiscontinuous Galerkin Approximations of Elliptic

Problems

Paola F. Antonietti

Department of Mathematics, University of Pavia

Joint work with B. Ayuso

Workshop ”Discontinuous Galerkin Methods”Bergamo, 24 February 2005

Paola F. Antonietti (University of Pavia) Schwarz Preconditioners for DG Methods Bergamo, 24 February 2005 1 / 27

Outline

1 Introduction

2 Unified Framework of DG Approximations

3 Additive and Multiplicative Non-Overlapping Schwarz Preconditioners

4 Theoretical AnalysisAdditive Schwarz PreconditionersMultiplicative Schwarz Preconditioners

5 Numerical Results

Features of DG Methods

√Wide range of PDE’s treated within the same unified framework

√Weak approximation of boundary conditions

√Flexibility in the mesh design & adaptivity strategies (hp)

• •

•• •

√non-matching grids (hanging nodes)

√non-uniform approximation degrees

√freedom in the choice of basis functions

X High number of degrees of freedom (dof)

X Large algebraic linear systems to solve (ill conditioned ≈ O(h−2))

Remedy

Develop optimal and efficient preconditioners

Features of DG Methods

√Wide range of PDE’s treated within the same unified framework

√Weak approximation of boundary conditions

√Flexibility in the mesh design & adaptivity strategies (hp)

• •

•• •

√non-matching grids (hanging nodes)

√non-uniform approximation degrees

√freedom in the choice of basis functions

X High number of degrees of freedom (dof)

X Large algebraic linear systems to solve (ill conditioned ≈ O(h−2))

Remedy

Develop optimal and efficient preconditioners

What is a Schwarz Method?

Alternating Schwarz method (1870):

90’s up to now: domain decompositiontechniques

Ω1 Ω2Γ1Γ2

Domain decomposition techniques for DG methods under development

Feng & Karakashian (SINUM, 2001),

Lasser & Toselli (Math. Comp., 2003),

Brenner & Wang (South Carolina Tech. Rep., 2004),

A. & Ayuso (M2AN, 2005, submitted)

The Basic Idea (Divide and Conquer)

Find uh ∈ Vh s.t. L(uh, vh) = F(vh), ∀ vh ∈ Vh

Decompose the original discrete problem into anumber of local subproblems

Solve the easier (and cheaper!) subproblems

Couple + coarse correction

A good preconditioner should be

Efficient: the condition number/iteration counts of the preconditionedsystem are improved

Scalable: the performance does not depend on the number of subdomains

Parallel: the algorithm should have a high level of parallelism

Cheap: the action of the preconditioner should be cheap to compute

The Basic Idea (Divide and Conquer)

Find uh ∈ Vh s.t. L(uh, vh) = F(vh), ∀ vh ∈ Vh

Decompose the original discrete problem into anumber of local subproblems

Solve the easier (and cheaper!) subproblems

Couple + coarse correction

A good preconditioner should be

Efficient: the condition number/iteration counts of the preconditionedsystem are improved

Scalable: the performance does not depend on the number of subdomains

Parallel: the algorithm should have a high level of parallelism

Cheap: the action of the preconditioner should be cheap to compute

Model Problem (Toy Problem)

Ω ⊂ Rd (d = 2, 3) bounded convex (polygonal), f ∈ L2(Ω)

−∆u = f in Ω

u = 0 on ∂Ω

Remark

More general boundary conditions (non-homogeneous Dirichlet and/orNeumann)

More general second order operators in divergence form (i.e., Darcy Law)

−∆u = f in Ω

u = 0 on ∂Ωσ = ∇u−−−−−−−→

σ = ∇u in Ω

−div(σ) = f in Ω

u = 0 on ∂Ω

Remark

−∆u = f in Ω

u = 0 on ∂Ωσ = ∇u−−−−−−−→

σ = ∇u in Ω

−div(σ) = f in Ω

u = 0 on ∂Ω

Remark

Starting Point: Partition, DG FE spaces, Trace Operators

Mesh: shape-regular partition Th with possible hanging nodes (h mesh size)

Local mesh size: h(x) = minhT+ , hT−, if x is in the interior of∂T+ ∩ ∂T−, h(x) = hT if x is in the interior of ∂T ∩ ∂Ω

Set of faces: E I ,E B sets of all internal/boundary faces and E = E I ∪ E B

Discontinuous FE spaces: `h ≥ 1 (M`h = P`, Q`)

Vh =v ∈ L2(Ω) : v |T ∈M`h(T ) ∀T ∈ Th

, Σh = [Vh]

Trace operators on internal faces: fix δ ∈ [0, 1], ∀ e ∈ E I shared by T±

vδ = δv+ + (1− δ)v− [[v ]] = v+n+ + v−n+

τδ = δτ+ + (1− δ)τ− [[τ ]] = τ+ · n+ + τ− · n+

Trace operators on boundary faces: for each e ∈ E B

vδ = v [[v ]] = vn τδ = τ [[τ ]] = τ · n

Vh =v ∈ L2(Ω) : v |T ∈M`h(T ) ∀T ∈ Th

, Σh = [Vh]

vδ = δv+ + (1− δ)v− [[v ]] = v+n+ + v−n+

τδ = δτ+ + (1− δ)τ− [[τ ]] = τ+ · n+ + τ− · n+

vδ = v [[v ]] = vn τδ = τ [[τ ]] = τ · n

Vh =v ∈ L2(Ω) : v |T ∈M`h(T ) ∀T ∈ Th

, Σh = [Vh]

vδ = δv+ + (1− δ)v− [[v ]] = v+n+ + v−n+

τδ = δτ+ + (1− δ)τ− [[τ ]] = τ+ · n+ + τ− · n+

vδ = v [[v ]] = vn τδ = τ [[τ ]] = τ · n

Unified DG Approximation [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001]

Find u ∈ Vh s.t. ADGh (u, v) = (f , v) ∀ v ∈ Vh

ADGh (u, v) =

∇hu · ∇hv +

[[u − u]] · ∇hv

∫E I

u − u [[∇hv ]]−∫

σ · [[v ]]−∫

[[σ]] v

Numerical Fluxes u = u(u) and σ = σ(σ, u)

Approximations of the traces of u and σ on the boundaries of the elements T

u = u(u) ≈ u∣∣∂Tσ = σ(u,σ) ≈ ∇u∣∣∂T

Different choices of u and σ determine all the DG methods.

Unified DG Approximation [Arnold, Brezzi, Cockburn & Marini, SINUM, 2001]

Find u ∈ Vh s.t. ADGh (u, v) = (f , v) ∀ v ∈ Vh

ADGh (u, v) =

∇hu · ∇hv +

[[u − u]] · ∇hv

∫E I

u − u [[∇hv ]]−∫

σ · [[v ]]−∫

[[σ]] v

Numerical Fluxes u = u(u) and σ = σ(σ, u)

Approximations of the traces of u and σ on the boundaries of the elements T

u = u(u) ≈ u∣∣∂Tσ = σ(u,σ) ≈ ∇u∣∣∂T

Different choices of u and σ determine all the DG methods.

Example: Interior Penalty Family Methods

2[[u]] · nT e ∈ E I

0 e ∈ E Bσ = ∇u − αe h

−1 [[u]] e ∈ E

AIPh (u, v) =

∫Ω∇hu · ∇hv −

∫E∇hu · [[v ]]

− (1− γ)

[[u]] · ∇hv −∑e∈E

∫eh−1 [[u]] · [[v ]]

γ = 0 SIPG method [Arnold, SINUM, 1982]

γ = 2 NIPG method [Riviere, Wheeler & Girault, Comp. Geosc.,1999]

γ = 1 IIPG method [Wheeler, Dawson & Sun, CMAME, 2004]

2[[u]] · nT e ∈ E I

0 e ∈ E Bσ = ∇u − αe h

−1 [[u]] e ∈ E

AIPh (u, v) =

∫Ω∇hu · ∇hv −

∫E∇hu · [[v ]]

− (1− γ)

[[u]] · ∇hv −∑e∈E

∫eh−1 [[u]] · [[v ]]

2[[u]] · nT e ∈ E I

0 e ∈ E Bσ = ∇u − αe h

−1 [[u]] e ∈ E

AIPh (u, v) =

∫Ω∇hu · ∇hv −

∫E∇hu · [[v ]]

− (1− γ)

[[u]] · ∇hv −∑e∈E

∫eh−1 [[u]] · [[v ]]

Numerical Fluxes: Summary Table

Method u(u) σ(σ, u)

SIPG u ∇hu − αeh−1 [[u]]BRMPS u ∇hu+ αe re([[u]])SIPG(δ) u(1−δ) ∇huδ − αeh−1 [[u]]

BO∗ u+ [[u]] · nT ∇huNIPG u+ [[u]] · nT ∇hu − αeh−1 [[u]]IIPG u+ 1/2 [[u]] · nT ∇hu − αeh−1 [[u]]

BR∗ u σBMMPR u σ+ αe re([[u]])LDG u − β · [[u]] σ+ β · [[σ]]− αeh−1 [[u]]

BZ∗ (u|T )|∂T −αeh−(2`+1) [[u]]BMMPR 2∗ (u|T )|∂T αeh−2` re([[u]])

Non-Overlapping Schwarz Preconditioners: Notation

Partitions:

TS = Ωi , i = 1, . . . ,Ns subdomain partition

TH coarse grid partition

Th fine grid partition (E the set of all faces) Ω1 Ω2

Ω3 Ω4

Skeleton:

Ei = e ∈ E : e ⊂ Ωi

Γij = e ∈ Ei : e ∈ ∂Ωi ∩ ∂Ωj

Γi = e ∈ Ei : e ∈ ∂Ωi \ ∂Ω

Γ = ∪Ns

i=1Γi

Ω1 Ω2

Ω3 Ω4

Partitions:

Ω3 Ω4

Skeleton:

Ei = e ∈ E : e ⊂ Ωi

Γij = e ∈ Ei : e ∈ ∂Ωi ∩ ∂Ωj

Γi = e ∈ Ei : e ∈ ∂Ωi \ ∂Ω

Γ = ∪Ns

i=1Γi

Ω1 Ω2

Ω3 Ω4

Partitions:

Ω3 Ω4

Skeleton:

Ei = e ∈ E : e ⊂ Ωi

Γij = e ∈ Ei : e ∈ ∂Ωi ∩ ∂Ωj

Γi = e ∈ Ei : e ∈ ∂Ωi \ ∂Ω

Γ = ∪Ns

i=1Γi

Ω1 Ω2

Ω3 Ω4

Partitions: TS ⊆ TH ⊆ Th

Ω3 Ω4

Skeleton:

Ei = e ∈ E : e ⊂ Ωi

Γij = e ∈ Ei : e ∈ ∂Ωi ∩ ∂Ωj

Γi = e ∈ Ei : e ∈ ∂Ωi \ ∂Ω

Γ = ∪Ns

i=1Γi

Ω1 Ω2

Ω3 Ω4

Partitions: TS ⊆ TH ⊆ Th

Ω3 Ω4

Skeleton:

Ei = e ∈ E : e ⊂ Ωi

Γij = e ∈ Ei : e ∈ ∂Ωi ∩ ∂Ωj

Γi = e ∈ Ei : e ∈ ∂Ωi \ ∂Ω

Γ = ∪Ns

i=1Γi

Ω1 Ω2

Ω3 Ω4

For each subdomain Ωi , i = 1, . . . ,Ns

Local Spaces: V ih =

v ∈ Vh : v ≡ 0 in Ω \ Ωi

h = [V ih]

Prolongation Operators: RTi : V i

h −→ Vh and DTi : Σi

h −→ Σh

Set to zero the degrees of freedom outside Ωi

Action on the interface Γi :

RTi vi =

−= 0

DTi τ i =

i τ i

= τ i

i τ i

−= τ i

For the coarse level

Coarse Space: 0 ≤ `H ≤ `h

VH = V 0h =

v ∈ L2(Ω) : v |D ∈M`H (D), ∀D ∈ TH

, ΣH = Σ0

h = [V 0h ]d

Prolongation Operators: RT0 : V 0

h −→ Vh and DT0 : Σ0

h −→ Σh.

v ∈ Vh : v ≡ 0 in Ω \ Ωi

h = [V ih]

h −→ Σh

RTi vi =

−= 0

DTi τ i =

i τ i

= τ i

i τ i

−= τ i

VH = V 0h =

v ∈ L2(Ω) : v |D ∈M`H (D), ∀D ∈ TH

, ΣH = Σ0

h = [V 0h ]d

h −→ Σh.

v ∈ Vh : v ≡ 0 in Ω \ Ωi

h = [V ih]

h −→ Σh

RTi vi =

−= 0

DTi τ i =

i τ i

= τ i

i τ i

−= τ i

VH = V 0h =

v ∈ L2(Ω) : v |D ∈M`H (D), ∀D ∈ TH

, ΣH = Σ0

h = [V 0h ]d

h −→ Σh.

v ∈ Vh : v ≡ 0 in Ω \ Ωi

h = [V ih]

h −→ Σh

RTi vi =

−= 0

DTi τ i =

i τ i

= τ i

i τ i

−= τ i

VH = V 0h =

v ∈ L2(Ω) : v |D ∈M`H (D), ∀D ∈ TH

, ΣH = Σ0

h = [V 0h ]d

h −→ Σh.

Non-Overlapping Schwarz Preconditioners: Local Solvers

Local solvers: −∆ui = Ri f in Ωi , ui = 0 on ∂Ωi , i = 1, . . . ,Ns ,

ADGi (ui , vi ) :=

∫Ωi

∇hui · ∇hvi +

[[ui − ui ]] · ∇hvi

∫E I

ui − ui [[∇hvi ]]−∫

σi · [[vi ]]−∫

[[σi ]] vi,

where ui and σi are the local numerical fluxes

Properties of the global and local numerical fluxes and bilinear forms

ui (ui ) = u(RTi ui ) σi (σi , ui ) = σ(DT

i σi ,RTi ui ) on e ∈ Ei

ADGi (ui , vi ) = ADG

h (RTi ui ,R

Ti vi ), ∀ ui , vi ∈ V i

ADGi (ui , vi ) :=

∫Ωi

∇hui · ∇hvi +

[[ui − ui ]] · ∇hvi

∫E I

ui − ui [[∇hvi ]]−∫

σi · [[vi ]]−∫

[[σi ]] vi,

h (RTi ui ,R

ADGi (ui , vi ) :=

∫Ωi

∇hui · ∇hvi +

[[ui − ui ]] · ∇hvi

∫E I

ui − ui [[∇hvi ]]−∫

σi · [[vi ]]−∫

[[σi ]] vi,

h (RTi ui ,R

Non-Overlapping Schwarz Preconditioners: Coarse Solver

Coarse Solver: for all u0, v0 ∈ V 0h

ADG0 (u0, v0) = ADG

h (RT0 u0,R

T0 v0)︸︷︷︸

Restriction of ADGh to V 0

Remark

ADG0 (u0, v0) = ADG

h (RT0 u0,R

T0 v0)6=ADG

H (u0, v0)

Conforming coarse solver is possible

ADG0 (u0, v0) = ADG

h (RT0 u0,R

T0 v0)︸︷︷︸

Remark

ADG0 (u0, v0) = ADG

h (RT0 u0,R

T0 v0)6=ADG

H (u0, v0)

ADG0 (u0, v0) = ADG

h (RT0 u0,R

T0 v0)︸︷︷︸

Remark

ADG0 (u0, v0) = ADG

h (RT0 u0,R

T0 v0)6=ADG

H (u0, v0)

Additive and Multiplicative Schwarz Preconditioners

Projection like operators: for i = 0, ...,Ns , we define Pi : Vh −→ Vh as

ADGh (Piu,RT

i vi ) = ADGh (u,RT

i vi ) ∀ vi ∈ V ih

ADDITIVE Schwarz Preconditioner:

Pad =Ns∑i=0

MULTIPLICATIVE Schwarz Preconditioner:

Pmu = I − (I − PNs ) . . . (I − P1)(I − P0) = I − Emu

Comments on symmetry of the operators Pad and Pmu

Pad is symmetric only for symmetric DG approximations

Pmu is non-symmetric for all DG approximations

Additive and Multiplicative Schwarz Preconditioners: Matrix Notation

Let A , Ai be the matrix representation of the bilinear forms ADGh and ADG

Let RTi , Ri be the matrix representation of the prolongation and restriction

operators RTi , Ri ,

Pi = RiA−1i RT

i A i = 0, . . . ,Ns

Pad =Ns∑i=0

(RiA−1i RT

Pmu = I− (I− ANs A−1Ns

A) . . . (I− R1A−11 A1A)(I− R0A

−10 A0A)

The preconditioned linear systems

Then, instead of solving the original problem Au = f we solve

Padu = g1 or Pmuu = g2

g1, g2 being suitable rhs.

Convergence Analysis: Following the Theory of Schwarz Methods

Stable Decomposition: a stable splitting can be found for the family ofsubspaces and the corresponding bilinear forms: ∃ C0 > 0 s.t.

Ns∑i=0

ADGi (ui , ui ) ≤ C 2

0ADGh (u, u) C 2

0 = O(Hh−1

)i = 0, . . . ,Ns

Strengthened Cauchy-Schwarz inequalities: ∃ 0 ≤ εij ≤ 1, i , j = 1, . . . ,Ns ,

|ADGh (RT

i ui ,RTj uj)| ≤ εijADG

h (RTi ui ,R

Ti ui )

1/2ADGh (RT

j uj ,RTj uj)

Local Stability: there exists ω ∈ (0, 2) s.t.

ADGh (RT

i ui ,RTi ui ) ≤ ωADG

i (RTi ui ,R

Ti ui ) 0 ≤ i ≤ Ns

Ns∑i=0

0ADGh (u, u) C 2

0 = O(Hh−1

)i = 0, . . . ,Ns

|ADGh (RT

h (RTi ui ,R

Ti ui )

1/2ADGh (RT

j uj ,RTj uj)

ADGh (RT

i (RTi ui ,R

Ti ui ) 0 ≤ i ≤ Ns

Ns∑i=0

0ADGh (u, u) C 2

0 = O(Hh−1

)i = 0, . . . ,Ns

|ADGh (RT

h (RTi ui ,R

Ti ui )

1/2ADGh (RT

j uj ,RTj uj)

ADGh (RT

i (RTi ui ,R

Ti ui ) 0 ≤ i ≤ Ns

Theory of Additive Schwarz Methods: Symmetric DG Approximations

For symmetric DG approximations Pad is symmetric.

Theorem [A. & Ayuso, M2AN, submitted]

Let ADGh be the bilinear form of one of the symmetric DG methods. Let Pad be

its additive Schwarz operator. Then,

κ(Pad) ≤ C (Nc)C20 = O

Nc is the maximum number of adjacent subdomains a given Ωi can have.

Theory of Additive Schwarz Methods: Symmetric DG Approximations

For symmetric DG approximations Pad is symmetric.

Let ADGh be the bilinear form of one of the symmetric DG methods. Let Pad be

its additive Schwarz operator. Then,

κ(Pad) ≤ C (Nc)C20 = O

Nc is the maximum number of adjacent subdomains a given Ωi can have.

Theory of Additive Schwarz Methods: Non-Symmetric DG Approximations

For non-symmetric DG approximations Pad is non-symmetric.GMRES error reduction factor [Eisentat, Elman & Schultz, SIAM, 1983]:

‖rn‖a ≤(

1− cp2

‖r0‖a

‖ · ‖a is the norm induced by the symmetric part a(·, ·) of ADGh (·, ·) and

cp(Pad) = infu 6=0

a(Padu, u)

a(u, u)Cp(Pad) = sup

a(Padu,Padu)

a(u, u)

Let ADGh be the bilinear form of one of the non-symmetric DG methods. Let Pad

be its additive Schwarz operator. If α∗ = mine∈E αe ≥ CHh−1, then

CC−20 ≤ cp(Pad) Cp(Pad) ≤ C (Nc)

Nc is the maximum number of adjacent subdomains a given Ωi can have andC 2

0 = O(Hh−1).

Theory of Multiplicative Schwarz Methods: Symmetric DG Approximations

Pmu = I − Emu Emu = (I − PNs ) . . . (I − P1)(I − P0)

Pmu is non-symmetric both for symmetric and non-symmetric DG approximations.Pi is symmetric for symmetric DG approximations.

‖Emu‖2a = sup

0 6=v∈Vh

a(Emuv ,Emuv)

a(v , v)

Theorem [A. & Ayuso, in preparation]

Let ADGh be the bilinear form of one of the symmetric DG methods. Let Pmu be

its multiplicative Schwarz operator. Then,

‖Emu‖2a = ‖I − Pmu‖2

a ≤ (1− θ), θ =2− ω

C 20 (2 + Nc)

, C 20 = O(Hh−1)

Nc being the maximum number of adjacent subdomains a given Ωi can have.Then ρ(Emu) < 1 and the multiplicative Schwarz method converges.

Theory of Multiplicative Schwarz Methods: Symmetric DG Approximations

Pmu = I − Emu Emu = (I − PNs ) . . . (I − P1)(I − P0)

Pmu is non-symmetric both for symmetric and non-symmetric DG approximations.Pi is symmetric for symmetric DG approximations.

‖Emu‖2a = sup

0 6=v∈Vh

a(Emuv ,Emuv)

a(v , v)

Let ADGh be the bilinear form of one of the symmetric DG methods. Let Pmu be

its multiplicative Schwarz operator. Then,

a ≤ (1− θ), θ =2− ω

C 20 (2 + Nc)

, C 20 = O(Hh−1)

Theory of Multiplicative Schwarz Methods: Non-Symmetric DG Approximations

Pmu is non-symmetric both for symmetric and non-symmetric DG approximations.Pi is non-symmetric for non-symmetric DG approximations.

‖Emu‖2a = sup

0 6=v∈Vh

a(Emuv ,Emuv)

a(v , v)

Let ADGh be the bilinear form of one of the non-symmetric DG methods. Let Pmu

be its multiplicative Schwarz operator. If the penalty parameterα∗ = mine∈E αe ≥ C (Hh−1)1/(Ns+1) Then,

a ≤ (1− θ), θ =C (Nc)

, C 20 = O(Hh−1)

Theory of Multiplicative Schwarz Methods: Non-Symmetric DG Approximations

Pmu is non-symmetric both for symmetric and non-symmetric DG approximations.Pi is non-symmetric for non-symmetric DG approximations.

‖Emu‖2a = sup

0 6=v∈Vh

a(Emuv ,Emuv)

a(v , v)

Let ADGh be the bilinear form of one of the non-symmetric DG methods. Let Pmu

be its multiplicative Schwarz operator. If the penalty parameterα∗ = mine∈E αe ≥ C (Hh−1)1/(Ns+1) Then,

a ≤ (1− θ), θ =C (Nc)

, C 20 = O(Hh−1)

Theory of Additive and Multiplicative Schwarz Methods: Summary Table

Type Prec Type DG Cond. Numb. Iter. Counts Rest.

Additive Symm O(Hh−1) O(√

Hh−1) -

Additive Non-Symm - O(√

Hh−1) α∗ > CHh−1

Multiplicative Symm - O(√

Hh−1) -

Multiplicative Non-Symm - O(√

Hh−1) α∗ > C(Hh−1)1/(Ns+1)

RemarkHypothesis on the penalty parameter α∗ is only technical and it is not required inpractice

Numerical Results

Ω = [0, 1]× [0, 1]

u(x , y) = sin(πx) sin(πy)

TS : Ωi ’s are squares (Ns = 4, 16)

αe = α,∀ e ∈ E

m successive global uniformrefinements of the initial grids,m = 0, 1, 2, 3.

Mesh sizes: Hn = H0/2m andhm = h0/2m, m = 0, 1, 2, 3

Figure: Initial coarse (top) and fine(bottom) refinements on a 4 subdomainpartition (mesh sizes H0 and h0)

Numerical Results: Non-Matching Grids

(a) Additive preconditioner, Ns = 4PPPPPH

h h0 h0/2 h0/4 h0/8

H0 31.39 65.88 137.24 277.76H0/2 6.32 32.81 67.12 137.05H0/4 - 6.36 32.99 67.06H0/8 - - 6.43 32.00

IT(Ah) 0.04e+5 0.17e+5 0.70e+5 2.79e+5

(b) Additive preconditioner, Ns = 16PPPPPH

h h0 h0/2 h0/4 h0/8

H0 29.33 63.32 133.88 272.87H0/2 6.10 31.47 65.47 135.81H0/4 - 6.36 32.84 66.89H0/8 - - 6.42 32.91

IT(Ah) 0.04e+5 0.17e+5 0.70e+5 2.79e+5

Table: SIPG method (α = 10): `h = 1, `H = 1

Numerical Results: Unstructured Triangular Grids

(a) Multiplicative preconditioner, Ns = 4PPPPPH

h h0 h0/2 h0/4 h0/8

H0 10 16 24 35H0/2 1 10 16 25H0/4 - 1 10 16H0/8 - - 1 10

IT(Ah) 41 73 137 x

(b) Multiplicative preconditioner, Ns = 16PPPPPH

h h0 h0/2 h0/4 h0/8

H0 11 16 24 35H0/2 1 10 17 25H0/4 - 1 10 16H0/8 - - 1 11

IT(Ah) 41 73 137 x

Table: NIPG method (α = 10): `h = 1, `H = 1

Numerical Results: Non-Matching Grids

(a) Additive preconditionerPPPPPH

hh0 h0/2 h0/4 h0/8

H0 30 46 73 111H0/2 15 33 52 80H0/4 - 16 35 54H0/8 - - 16 36IT(Ah) 113 205 391 x

(b) Multiplicative preconditionerPPPPPH

hh0 h0/2 h0/4 h0/8

H0 13 22 34 51H0/2 1 12 20 31H0/4 - 1 12 19H0/8 - - 1 11IT(Ah) 113 205 391 x

Table: SIPG method (α = 10): `h = 1, `H = 1, Ns = 16

(a) Additive preconditioner, α = 10PPPPPH

hh0 h0/2 h0/4 h0/8

H0 30 46 70 ∗106H0/2 15 33 52 78H0/4 - 15 34 51H0/8 - - 15 34IT(Ah) 122 210 388 x

(b) Additive preconditioner, α = 2PPPPPH

hh0 h0/2 h0/4 h0/8

H0 18 25 35 49H0/2 ∗13 20 26 35H0/4 - ∗15 20 26H0/8 - - ∗15 20IT(Ah) 65 107 198 x

Table: NIPG method: `h = 1, `H = 1, Ns = 16

Works in Preparation and Future Works

Multiplicative Schwarz methods [A. & Ayuso, in preparation]

Schwarz methods for convection-diffusion problems [A. & Suli, in progress]

Multigrid methods [A., Ayuso & Zikatanov, in progress]

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