Towards Multigrid Methods - TUM · The Multigrid V-Cycle Crucial idea: recursive call of...

Scientific Computing II

Towards Multigrid Methods

Michael BaderTechnical University of Munich

Summer 2018

Part I

Multigrid Methods

Fundamental Multigrid IdeasNested IterationCoarse-Grid Correction – A Two-Grid Method

Multigrid SchemesMultigrid V-CycleMultigrid W-CycleFull Multigrid V-Cycle

Computational EffortCosts per IterationSpeed of Convergence

Michael Bader | Scientific Computing II | Towards Multigrid Methods | Summer 2018 2

Multigrid Idea No. 1

Observation and convergence analysis show:• “high-frequency error” is relative to mesh size• on a sufficiently coarse grid, even very low frequencies can be

“high-frequency”(if the mesh size is big)

“Multigrid” idea:• use multiple grids to solve the system of equations• hope that on each grid, a certain range of error frequencies will be

reduced efficiently

Nested Iteration

Solve the problem on a coarser grid:• will be comparably (very) fast• can give us a good initial guess:• leads to “poor man’s multigrid”: nested iteration

Algorithm:1. Start on a very coarse grid with mesh size h = h0;

guess an initial solution xh

2. Iterate over Ahxh = bh using relaxation method⇒ approximate solution xh

3. interpolate the solution xh to a finer grid Ωh/2

4. proceed with step 2 (now with mesh size h := h/2) using interpolatedxh/2 as initial solution

Multigrid Idea No. 2

Observation for nested iteration:• error in interpolated initial guess also includes low frequencies• relaxation therefore still slow• can we go “back” to a coarser grid later in the algorithm?

Idea No. 2: use the residual equation⇒ coarse-grid correction:

• solve Ae = r on a coarser grid• leads to an approximation of the error e• add this approximation to the fine-grid solution

A Two-Grid Method

Algorithm:1. relaxation/smoothing on the fine level system⇒ solution xh

2. compute the residual rh = bh − Ahxh

3. restriction of rh to the coarse grid ΩH

4. compute a solution to AHeH = rH

5. interpolate the coarse grid solution eH to the fine grid Ωh

6. add the resulting correction to xh

7. again, relaxation/smoothing on the fine grid

Correction Scheme – Components

• smoother:reduce the high-frequency error components, and get a smooth error

• restriction:transfer residual from fine grid to coarse grid, for example by

– injection– (full) weighting

• coarse grid equation:(acts as) discretisation of the PDE on the coarse grid

• interpolation:transfer coarse grid solution/correction from coarse grid to fine grid

The Multigrid V-Cycle

Crucial idea: recursive call of Two-/Multigrid solver on coarse grids

Algorithm:1. pre-smoothing on the fine level system⇒ solution xl

2. compute the residual rl = bl − Alxl

3. restriction of rl to the coarse grid Ωl−1

4. solve coarse grid system Al−1el−1 = rl−1by a recursive call to the V-cycle algorithm

5. interpolate the coarse grid solution el−1 to the fine grid Ωl

6. add the resulting correction to xl

7. post-smoothing on the fine grid

The W-Cycle

• perform two coarse grid correction steps instead of one

Ω4h Ω4h

Ω2h Ω2h

Ωh Ωh

Ω8h Ω8h

Ω4h Ω4h Ω4h

Ω8h Ω8h

Ω4h Ω4h Ω4h

Ω2h Ω2h Ω2h

Ωh Ωh

AAU AAU AAU

(V-cycle and W-cycle)

• more expensive• useful in situations where the coarse grid correction is not very accurate

The Full Multigrid V-Cycle (FMV- or F-Cycle)

Recursive algorithm:• combines nested iteration and V-cycle• (recursively!) perform an FMV-cycle on the next coarser grid to get a

good initial solution• interpolate this initial guess to the current grid• perform a V-cycle to improve the solution

Ω8h Ω8h

Ω4hΩ4h

Ω4h Ω4h

Ω2h Ω2h

Ω4h Ω4h

Ω2h Ω2h

Ωh Ωh

V-Cycle – Computational Costs

Further algorithmic details:• on the coarsest grid: direct solution;

but: number of unknowns small, ideally O(1)

• number of smoothing steps is typically very small (1 or 2)and must not depend on problem size

Computational Costs (storage and computing time):• 1D: c · n + c · n/2 + c · n/4 + . . . ≤ 2c · n• 2D: c · n + c · n/4 + c · n/16 + . . . ≤ 4/3c · n• 3D: c · n + c · n/8 + c · n/64 + . . . ≤ 8/7c · n• overall costs are dominated by the costs of the finest grid

(n the number of grid points on the finest grid: typicall n = h−D)

Thus: runtime O(n) per iteration, but how many iterations necessary?

Speed of Convergence

• fastest method around (if all components are chosen carefully), but:best-possible convergence often hard to obtain

• “textbook multigrid efficiency”:∥∥∥e(m+1)∥∥∥ ≤ γ ∥∥∥e(m)

∥∥∥ ,where convergence rate γ < 1 (esp. γ 1) is independent of the numberof unknowns⇒ constant number of multigrid steps to obtain a given number of digits⇒ overall computational work increases only linearly with the number of

unknowns• see exercises: analysis of two-grid convergence

Speed of Convergence (2)For the “Model Problem” (i.e., Poisson Problem):

• O(n) to solve up to “reduce error by a factor of . . . ” (10−8, e.g.)• wanted: solve to the “level of truncation”→ depending on discretisation error; for example O(h2)

• O(n) up to “level of truncation” achieved by FMV-Cycle;ideal case: 1 cycle; after each V-cycle, the “level of truncation” error isachieved on that grid on that grid

For Other Problems:• OK for strongly elliptic problems• multigrid variants for non-linear problems, parabolic/hyperbolic, . . .• every component may fail, leading to slow or no convergence:

smoother, interpolation/restriction, coarse-grid operator• achieving “textbook efficiency” usually a demanding task

Part II

Components of Multigrid Methods

InterpolationRestrictionCoarse Grid OperatorSmoothers

Interpolation (aka “Prolongation”)

For Poisson problem:• (bi-)linear interpolation:

in 1D: resembles homogeneous (f = 0) solution• constant (in general too small approximation order):

sometimes used for cell-based coarsening (unknowns located in cellcenters)

• quadratic, cubic, etc.:often too costly, more smoothing steps are cheaper and can eliminate thedisadvantage of a lower-order interpolation

• but: in FMV-cycle interpolation to finer grid (after a completed V-cycle)should be higher-order (to limit the introduced error)

Interpolation – Matrix Notation

For linear interpolation (1D):

12 0 01 0 012

0 1 00 1

0 0 10 0 1

12 (0 + x1)

x112 (x1 + x2)

x212 (x2 + x3)

x312 (x3 + 0)

Notation: Ih2hx2h = xh or Ph

2hx2h = xh

Note: disregards boundary values (here: 0-Dirichlet condition assumed)

Interpolation – Convection-DiffusionExample problem: 1D convection-diffusion equation

−εuxx + cux = f , 0 < ε c

Operator-dependent Interpolation:• consider homogeneous problem (f = 0)

with Dirichlet boundaries: u(0) = 1, u(1) = 0• exact solution for this case:

u(x) =1

1− ec/ε

(ecx/ε − ec/ε

)= 1− 1− ecx/ε

1− ec/ε

• interpolate at x = 12 : u

)= 1− 1−ec/2ε

1−ec/ε =: 1− 1−z1−z2 ;

for large z = ec

2ε , we have u( 1

)≈ 1− 1

z ≈ 1• thus: linear interpolation inappropriate (and leads to slow convergence)→ interpolation should be operator-dependent

RestrictionFor Poisson problem:

• “injection”: pick values at corresp. coarse grid points• “full weighting” = transpose of bilinear interpolation (safer, more robust

convergence), see illustration below for the 1D case

1/2 1/21/2 1/2

linear interpolation

1/21/2

f ll i hti

11/2 full weighting

Restriction – Matrix Notation

For full weighting (1D):

12 1 1

2 0 0 0 00 0 1

2 1 12 0 0

0 0 0 0 12 1 1

x1x2x3x4x5x6x7

12 (x1 + 2x2 + x3)12 (x3 + 2x4 + x5)12 (x5 + 2x6 + x7)

Notation: I2hh xh = x2h or R2h

h xh = x2h

Coarse Grid Operator

Two main options:1. discretise PDE on grid Ωh to obtain Ah

2. “Galerkin approach”: A2h := R2hh AhPh

2h→ compare effect on vector x2h:

A2hx2h := R2hh AhPh

→ evaluate from right to left:• interpolate x2h to xh := Ph

2hx2h• apply fine-grid operator Ah to interpolated xh• restrict resulting matrix-vector product to Ω2h

Exercise:• Compute A2h := R2h

h AhPh2h for Ah := 1

h2 tridiag(−1,2,−1)

A Matrix-Oriented View of Coarse-Grid Correction

1. given a system of equations Ahxh = bh

2. pre-smoothing leads to approximate solution x (i)h

and resp. error xh = x (i)h + e(i)

3. compute residual r (i)h = bh − Ahx (i)h ;

respective residual equation: Ahe(i)h = r (i)h

4. restriction of residual equation: R2hh Ahe(i)

h = R2hh r (i)h

5. approximate error on coarse grid: e(i)h ≈ Ph

2he(i)2h

→ leads to Galerkin coarsening R2hh AhPh

2he(i)2h = R2h

h r (i)h

6. compute error e(i)2h from (R2h

h AhPh2h)e(i)

2h = A2he(i)2h = b2h := R2h

h r (i)h

7. interpolate coarse-grid error, e(i)h ≈ Ph

2he(i)2h ,

and apply as coarse-grid correction: x (i+1)h = x (i)

h + Ph2he(i)

8. post-smoothing on approximate solution x (i+1)h

Galerkin Coarse Grid Operator: A2h := R2hh AhPh

• assume linear Interpolation and full weighting restriction:

R2hh =

12 1 1

2 0 0 0 00 0 1

2 1 12 0 0

0 0 0 0 12 1 1

Ph2h = (R2h

• for Poisson equation with stencil 1h2 [−1 2 − 1] (see exercises)

→ coarse-grid stencil reproduces: 1(2h)2 [−1 2 − 1]

• for pure convection and central differencing, stencil 1h [1 0 − 1],

→ coarse-grid stencil reproduces: 12h [1 0 − 1] (check!)

→ but leads to unstable discretisation• question: result for upwind discretisation: stencil 1

h [−1 1 0]?

Galerkin Coarsening and Convection

• coarsening with linear Interpolation, full weighting restriction,and upwind stencil 1

h [−1 1 0]

• leads to coarse-grid stencil 1h

[− 3

]→ not a diagonal-dominant matrix→ unstable discretisation

• remedy: use “downwind” interpolation and “upwind” restriction:

R2hh =

1 1 0 0 0 0 00 0 1 1 0 0 00 0 0 0 1 1 0

Ph2h =

0 0 01 0 01 0 00 1 00 1 00 0 10 0 1

• result: upwind discretisation stencil reproduced: 1

2h [−1 1 0]!

Matrix-Dependent Interpolation/Restriction• assume general 1D 3-point discretisation stencil: [sl sc sr ]

where sc = −(sl + sr ) > 0• use the following 3-point interpolation stencil:

]− sl

sc1 − sr

Ph2h = (R2h

h )T , thus:

R2hh =

−slsc

1 − srsc

0 0 0 00 0 − sl

sc1 − sr

0 0 0 0 − slsc

1 − srsc

• Galerkin coarsening A2h := R2h

h AhPh2h leads to stencil

[−s2

l s2l + s2

r − s2r]

→ check this!

• remains diagonal dominant→ stable coarse-grid discretisation• try as exercise:

compute coarse-grid operator for convection-diffusion equation;compare operator- and matrix-dependent interpolation/restriction

Matrix-Dependent Interpolation/Restriction (2)• take a closer look at the matrix multiplication R2h

. . .. . . 0 0 0 0

· · · 0 − slsc

1 − srsc

0 · · ·

0 0 0 0. . .

. . .. . .

0 sl sc sr 0 · · ·· · · 0 sl sc sr 0 · · ·

· · · 0 sl sc sr 0

. . .. . .

• equivalent to performing row operations as in Gaussian elimination:

. . .. . .

0 sl sc sr 0 · · ·· · · 0 sl sc sr 0 · · ·

· · · 0 sl sc sr 0

. . .. . .

·(− sl

·1·(− sr

• red entries become 0⇒ coarse-grid unknown no longer depends on fine-grid unknowns• similar for multiplication AhPh

2h, but with column operations

Smoothers

Efficient smoothers for the Poisson problem (see tutorials):• Gauss-Seidel• red-black Gauss-Seidel• damped (ω = 2

3 ) Jacobi→ why ω = 2

3 ?→ examine smoothing factors (dependent on wave number k )

How about . . .• Jacobi (non-weighted)?→ does not work (zig-zag pattern prevents smoothing)

• SOR?→ typically does not work well for Poisson model problem

(does not smooth high frequencies efficiently)→ can help for other problems using a tailored ω

Smoothers – Anisotropic ProblemsModel problems and observation:

• anisotropic Poisson equation: uxx + εuyy = f with ε 1(or similar: ε 1)

• Poisson equation on stretched grids:

x(ui+1,j − 2ui,j + ui−1,j ) +

y(ui,j+1 − 2ui,j + ui,j−1) = fij

with hx hy (or similar: hx hy )• Strong dependency in x-direction, weak dependency in y -direction

(or vice versa)• Good smoothing of the error only in x-direction

(or in y -direction):

ui,j =1

2h2x + 2h2

(−h2

xh2y fij + h2

y (ui+1,j + ui−1,j )− h2x (ui,j+1 + ui,j−1)

Smoothers – Anisotropic Problems (2)

Semi-Coarsening:• situation on coarser grid for example Hx = 2hx , Hy = hy :

x(ui+1,j − 2ui,j + ui−1,j ) +

y(ui,j+1 − 2ui,j + ui,j−1) = fij

• thus: anisotropy has weakened on the semi-coarsened grid

Line Smoothers:• perform a column-wise (or row-wise) Jacobi/Gauss-Seidel relaxation→ solve each column (or row) simultaneously:

u(n+1)i−1,j − (2 + 2ε)u(n+1)

ij + u(n+1)i+1,j = h2fij − ε(u(n)

i,j−1 + u(n)i,j+1)

• use direct, tridiagonal solver for each “line” (i.e., row or column)

Smoothers – Convection-Diffusion

• example: 1D Convection-Diffusion equation

−εuxx + ux = f , ε 1

• “upwind discretisaton”:

− ε

h2 (un−1 − 2un + un+1) +1h

(un − un−1) = fn

• (weighted) Jacobi and red-black Gauss-Seidel?→ no smoothing, basically updates one grid point per iteration

• Gauss-Seidel (relaxation from “left to right”)?→ almost an exact solver

Smoothers – More Complicated Situations

Problems:• anisotropic Poisson with space-dependent ε = ε(x , y),

or more general:−∇

(D(x , y)∇u(x , y)

)= f (x , y)

• convection-diffusion with variable convection:

−εuxx + v(x)ux = f − ε∆u + v(x , y)∇u(x , y) = f (x , y)

Approaches for Smoothing:• alternating line smoothers, “plane smoothers” in 3D• “Zebra” line smoothers (similar to red-black-GS)• Gauss-Seidel smoothing in “downwind” order→ difficult to do for complicated flows in 2D and 3D

Part III

Finite Elements, Hierarchical Bases,and Multilevel Methods

FEM Main IngredientsTest and Shape Functions1D Poisson Equation and Nodal Basis

Hierarchical Bases and Generating SystemsHierarchical BasisFEM with Hierarchical vs. Nodal BasesHierarchical Basis TransformationFEM and Hierarchical Basis Transform

Generating Systems and MultigridHierarchical Generating System and FEM

Remember: Finite Elements – Main Ingredients

1. solve weak form of PDE to reduce regularity properties

u′′ = f −→∫

v ′u′ dx =

∫vf dx

→ allows additional weak solutions2. compute a function as numerical solution→ search in a function space Wh:

uh =∑

ujϕj (x), spanϕ1, . . . , ϕJ = Wh

3. find weak solutions of simple form:for example piecewise linear functions and choose basis functions withlocal support (“hat functions”)→ leads to system of linear equations

Test and Shape Functions

• consider a general PDE Lu = f on some domain Ω

• search for solution functions uh of the form

uh =∑

ujϕj (x)

the ϕj (x) are typically called shape or ansatz functions• the basis functions ϕj (x) build a vector space

(i.e., a function space) Wh

spanϕ1, . . . , ϕJ = Wh

• insert into weak formulation∫vL(∑

ujϕj (x)

∫vf dx ∀v ∈ V

Test and Shape Functions (2)• choose a basis ψi of the test space Vh

typically defined on some discretisation grid Ωh• then: if all basis functions ψi satisfy∫

ψi (x)L(∑

ujϕj (x)

∫ψi (x)f (x) dx ∀ψi

then all v ∈ Vh satisfy the equation• the ψi are therefore often called test functions• we obtain a system of equations for unknowns uj :

one equation for each test function ψi• Vh is often chosen to be identical to Wh (Ritz-Galerkin method)→ we then have as many equations as unknowns

• leads to system of linear equations Au = b where

(Au)i :=∑

∫ψi (x)Lϕj (x) dx︸︷︷︸

∫ψi (x)f (x) dx =: bi

Example: Nodal Basis

ϕi (x) :=

1h (x − xi−1) xi−1 < x < xi1h (xi+1 − x) xi < x < xi+1

0 otherwise

0,60,4 0,80,20

Example Problem: 1D Poisson

• in 1D: −u′′(x) = f (x) on Ω = (0,1),hom. Dirichlet boundary cond.: u(0) = u(1) = 0

• weak form: ∫ 1

0v ′(x) · u′(x) dx =

0v(x)f (x) dx ∀v

• grid points xi = ih, (for i = 1, . . . ,n − 1); mesh size h = 1/n• Vh = Wh: piecewise linear functions (on intervals [xi , xi+1])• leads to stiffness matrix:

2 −1

−1 2. . .

. . . . . . −1−1 2

Hierarchical Basis

• hat functions with multi-level resolution

0,60,4 0,80,20

• FEM solution identical for hierarchical and nodal basis(same function space!)

• known from Scientific Computig I:diagonal stiffness matrix for 1D Poisson!

Hierarchical vs. Nodal Basis

h3=2-3xi

f(x)u(x)=iii(x)u(x)

h3=2-3xi

3u(x)=iii(x)

Hierarchical Basis and Multigrid

• What happens, if we use FEM on hat function bases with differentresolutions?

• Define “mother of all hat functions”

φ(x) := max1− |x |,0

• consider mesh size hn = 2−n and grid points xn,i = i · hn

• nodal basis then Φn := φn,i ,0 ≤ i ≤ 2n with

φn,i (x) := φ

(x − xn,i

)• hierarchical basis combines Φn := φn,i , i = 1,3, . . . ,2n − 1 (only odd

indices) and defines basis as

Ψn :=n⋃

Hierarchical Basis Transformation(or: How to represent functions on a coarser grid?)

0,60,4 0,80,20

−→

0,60,4 0,80,20

• represent hat functions φn−1,i (x) via fine-level functions φn,j (x)

φn−1,i (x) = 12φn,2i−1(x) + φn,2i (x) + 1

2φn,2i+1(x)

• hierarchical-basis transformation as matrix-vector product:ψn,i−1(x)ψn,i (x)ψn,i+1(x)

φn,2i−1(x)φn−1,i (x)φn,2i+1(x)

1 0 012 1 1

20 0 1

φn,2i−1(x)φn,2i (x)φn,2i+1(x)

Hierarchical Basis Transformation (2)

Level-by-level algorithm for hierarchical transform:

.x2x1 x3 x5 x7

Φ4Φ2 Φ6Φ1 Φ3 Φ5 Φ7

−→

.x2x1 x3 x5 x7

Ψ2 Ψ6Ψ1 Ψ3 Ψ5 Ψ7

.x2x1 x3 x5 x7

Ψ2 Ψ6Ψ1 Ψ3 Ψ5 Ψ7

−→

.x2x1 x3 x5 x7

Ψ2 Ψ6Ψ1 Ψ3 Ψ5 Ψ7

Remark: allows to implement transform in O(N)

Hierarchical Basis Transformation (3)

• hierarchical basis transformation: ψn,i (x) =∑

jHi,jφn,j (x)

• transform can be written as matrix-vector product: ~ψn = Hn~φn

• step-wise transform from each grid level to the next similar to

H(2)3 =

1 0 0 0 0 0 012 1 1

2 0 0 0 00 0 1 0 0 0 00 0 1

2 1 12 0 0

0 0 0 0 1 0 00 0 0 0 1

2 1 12

0 0 0 0 0 0 1

cmp. restriction operator!

• Hn then a sequence of level-to-next-level transforms:

Hn = H(1)n H(2)

n . . . H(n−2)n H(n−1)

.x2x1 x3 x5 x7

Φ4Φ2 Φ6Φ1 Φ3 Φ5 Φ7

−→

.x2x1 x3 x5 x7

Ψ2 Ψ6Ψ1 Ψ3 Ψ5 Ψ7

Hierarchical Coordinate Transformation

• consider function f (x) ≈∑

iaiψn,i (x) represented via hier. basis

• wanted: corresponding representation in nodal basis∑k

bkφn,k (x) =∑

aiψn,i (x) ≈ f (x)

• with ψn,i (x) =∑

jHi,jφn,j (x) we obtain

bkφn,k (x) =∑

Hi,jφn,j (x) =∑

aiHi,jφn,j (x)

• compare coordinates (identify indices j and k ) and get

bj =∑

Hi,jai =∑

j,i ai

• written in vector notation: b = HT a

FEM and Hierarchical Basis Transform

• FEM discretisation with hierarchical test and shape functions:∫ψi (x)L

ujψj (x)

∫ψi (x)f (x) dx ∀ψi

• leads to respective stiffness matrix AHBi,j :∫

ψi (x)L(∑

ujψj (x)

∫ψi (x)Lψj (x) dx =

ujAHBi,j

• vs. stiffness matrix with nodal basis as shape functions:∫ψi (x)L

vjφj (x)

∫ψi (x)Lφj (x) dx =

vjA∗i,j

• Note:∑

j ujAHBi,j and

∑j vjA∗i,j are both equal to

∫ψi (x)f (x) dx

• Therefore: (AHBu)i =∑

j ujAHBi,j =

∑j vjA∗i,j = (A∗v)i and v = HT u

FEM and Hierarchical Basis Transform (2)

• status: FEM with hierarchical test and nodal shape functions∫ψi (x)L

vjφj (x)

∫ψi (x)f (x) dx

• represent test functions via nodal basis:∫ ∑k

Hi,kφk (x)L(∑

vjφj (x)

∫ ∑k

Hi,kφk (x)f (x) dx

∫φk (x)L

vjφj (x)

∫φk (x)f (x) dx

• leads to new system of equations: HANB v = H bNB

where ANB and bNB stem from nodal-basis FEM discretisation!• with v = HT u we obtain H ANB HT u = H b as system of equations, thus:

AHB = H ANB HT ( Galerkin coarsening)

FEM and Hierarchical Basis Transform – Summary

• in general: FEM with nodal test and shape functions∫φi (x)L

vjφj (x)

∫φi (x)f (x) dx ANBuNB = bNB

changed to different test and shape functions:∫ψi (x)L

vjψj (x)

∫ψi (x)f (x) dx A??u?? = b??

• change from nodal to hierarchical basis:AHBuHB = H ANB HT uHB = H bNB = bHB

• change from nodal to coarser-level nodal basis:ACuC = R ANB RT uC = R bNB = RC

• Note: R,RT as in Galerkin coarsening; strongly related to H,HT

Hierarchical Generating System

x2,1 x2,3

x3,1 x3,3 x3,5 x3,7

Φ2,1 Φ2,3

Φ3,1 Φ3,3 Φ3,5 Φ3,7

x2,1 x2,3

x3,1 x3,3 x3,5 x3,7

x3,2 x3,6x3,4

• hierarchical basis combines Φn := φn,i , i = 1,3, . . . ,2n − 1 (only oddindices)

• generating system combines nodal bases of all levels:n⋃

l=1Φl

FEM and Hierarchical Generating Systems• solution function uh represented as

uh =n∑

2l−1∑j=1

vl,jφl,j (x)

→ non-unique “multi-grid” representation!• FEM discretisation with test and shape functions from hierarchical

generating system:∫φk,i (x)L

vl,jφl,j (x)

∫φk,i (x)f (x) dx ∀φk,i ∈

n⋃k=1

• assume order of test/shape functions:

φn,1, . . . , φn,2n−1, φn−1,1, . . . , φn−1,2n−1−1, . . . , φ2,1, φ2,2, φ2,3, φ1,1

• leads to (linearly dependent!) system of linear equations:

AGSuh = bGS

Multi-Level System of Equations• system matrix AGS given as (for example)Ah A2h

h A4hh

Ah2h A2h A4h

Ah4h A2h

4h A4h

• Ah, A2h, and A4h are the nodal-basis stiffness matrices for resolution h,

2h, and 4h• consider submatrix Ah

2h → computed as∫φ2h,i (x)L

vh,jφh,j (x)

∫φ2h,i (x)f (x) dx ∀φ2h,i ∈ Φ2h

• test functions φ2h,i (x) = 12φh,2i−1(x) + φh,2i (x) + 1

2φh,2i+1(x)

• not a hierarchical transform, but a restriction operation R2hh :

~φ2h = R2hh~φh thus: Ah

2h = R2hh Ah

Multi-Level System of Equations (2)

• system of linear equations AGSvGS = bGS thus given as Ah AhPh2h AhPh

R2hh Ah A2h A2hP2h

R4hh Ah R4h

2hA2h A4h

v2hv4h

h bhR4h

with restriction and prolongation operators: R = PT

• matrix is singular(!) due to obviously linearly dependent rows• however: all solutions lead to same piecewise linear function• result of a relaxation method on this system of equations?

symmetric Gauss-Seidel sweep corresponds to one multigridV-Cycle! (Griebel, 1994)

Symmetric Gauss-Seidel on AGSvGS = bGS

• pick out second block row of the system AGSvGS = bGS:

R2hh Ahvh + A2hv2h + A2hP2h

4h v4h = R2hh bh

• Gauss-Seidel→ thus solve for unknowns in v2h:

A2hv2h + A2hP2h4h v (old)

4h = R2hh bh − R2h

h Ahv (new)h = R2h

(bh − Ahv (new)

)• observation #1: relaxation works on restricted fine-grid residual

(bh − Ahv (new)

)• observation #2: relaxation considers prolongated coarse-grid correction

from previous iteration: A2hP2h4h v (old)

⇒ FEM on hierarchical generating systems matches V-Cycle Multigridwith Galerkin coarsening

Literature/References – Multigrid

• Briggs, Henson, McCormick: A Multigrid Tutorial (2nd ed.), SIAM, 2000.• Trottenberg, Oosterlee, Schuller: Multigrid, Elsevier, 2001.• Shapira: Matrix-Based Multigrid: Theory and Applications, Springer,

2008.• Hackbusch: Iterative Solution of Large Sparse Systems of Equations,

Springer 1993.• Brandt, Livne: Multigrid Techniques: 1984 Guide with Applications to

Fluid Dynamics, Revised Edition, SIAM• M. Griebel: Multilevelmethoden als Iterationsverfahren uber

Erzeugendensystemen, Teubner Skripten zur Numerik, 1994M. Griebel: Multilevel algorithms considered as iterative methods onsemidefinite systems, SIAM Int. J. Sci. Stat. Comput. 15(3), 1994.

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