Finite Elements Methodsfor Wave Equations

Workshop on Unstructured Meshesin Dynamical Spacetimes

Gerhard ZumbuschInstitut für Angewandte MathematikFriedrich-Schiller-Universität Jena

Galerkin Method - Grids

Consistent discrete schemes for•Unstructured grids•Adaptive grid refinement•Higher order approximation

•Spacetime grid•Local time-stepping

scalar, linear wave equation

Finite Differences in spacetime

leap-frog time stepping

titi+1

ti-1

t

x

Galerkin Method – 1D

jdxfdxa jjii

i ∀⋅=⋅∫ ∫∑1

0

1

0

'' φφφ

iiii f

huuu

=−+− +−

211 2

0)1()0(,'' ===− uufu

)1,0('' 10

1

0

1

0

Hvvdxfdxvu ∈∀⋅=⋅∫ ∫

Strong formulation, Finite Differences

Weak formulation

Galerkin schemeFinite Element Method

Pseudo-Spectral (Galerkin/ p) Method

e.g. piecewise linear φ

e.g. polynomial φ

FEM semidiscrete in space

piecewise linear φ in space, continuous

~ FD scheme in space

solve system M->choose numerical quadrature such that M is diagonal

DG semidiscrete in time

piecewise constant φ in time, discontinuous

=Euler scheme in time

FEM in space, DG in time

piecewise linear φ in space, continuous

piecewise constant φ in time, discontinuous

interior penalty DG semidiscrete in space

piecewise linear φ in space, discontinuous

Penalty parameters c

symmetric/unsymmetric IPDG

interior penalty DG semidiscrete in space (2)

x2

x1

M block-diagonalSolve system M->choose φ such that M is diagonal

piecewise linear φ in space, discontinuous

Space-time Galerkin schemes

New: FEM in spacetime

piecewise linear φ in spacetime, continuous

Solve system

Spacetime as Petrov-Galerkin FEM

1000 )(,)(0

uuuuu

ttt ===−

==

vxvuvuvua tt ∀=∇⋅∇+−= ∫ 0d)(),(M

0)(,0)( ende0 == == ttt vv

scalar wave equation

Petrov-Galerkin:ansatz u & test v differ

bi-linear test function v

titi+1

ti-1titi+1

ti-1

bi-linear ansatz function u

t

x

t

x

time stepping

New: interior penalty DG in spacetime

titi+1

ti-1

t

x

piecewise linear φ in spacetime, discontinuous

New: interior penalty DG in spacetime(2)

titi+1

ti-1

t

x

Solve block-diagonal system

Space-time meshes

Well posed discrete system

time

Cartesian mesh:# ansatz functions= # test functions

unstructured mesh:# ansatz functions= # test functions

CFL condition/ stabilityfor unstructured meshes?

space

time

Cartesian mesh: Δt < C Δx

Simplex: one time edged space edges

or: no time edged+1 space edges

Explicit schemefor unstructured meshes?

time

Cartesian mesh:nodes in ascending time ordertime slice by time slice

Uniform triangular grid:node by node

13 14 15 169 10 11 125 6 7 81 2 3 4

Simplex:node at begin of time edge first

1

2

3

space

time

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.93

2.4

4

5.5

7

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

explicit

CLF with spatial mesh refinement+local time stepping

Local time-stepping

No time edge

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

12345

678

910

111213141516

171819

20

21 22 23 24 25 26 27

grid

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.0

0.2

0.4

0.6

0.8

1.0

0

1.6

3.3

4.9

6.5

Refined spacetime meshes

space

time

Interpolation/hanging node

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

CLF with local space-time refinement

Refined space-time meshes

Einstein‘s equations

Einstein equation

metric

curvature

connection

Analogy: Maxwell equation

Lorenz gauge condition

Vector potential

Maxwell equation in Lorenz gauge

ill posed Cauchy problem

principal part

in harmonic gauge:symmetric hyperbolic

strong formulation

find metric gμν such that:ill posed Cauchy problem

harmonic gauge condition

Einstein-Hilbert action

weak formulation2nd derivatives

weak formulation

principal part

weakformulation

1st derivatives

linearized equation

Minkowski metric,flat background

nonlinear space-time FEM

nonlinear Symmetric InteriorPenalty Discontinuous Galerkin

titi+1

ti-1

t

x

Traveling wave 1+1

0 100 200 300 400 500 600 700 800 900 1000-5

10

-410

-310

-210

-110

010

110

linear wave in 1+1 dimensions, FD

time

max

erro

r

25

50

100

200

0 100 200 300 400 500 600 700 800 900 1000-4

10

-310

-210

-110

010

110

linear wave in 1+1 dimensions, SIPDG

time

max

erro

r

25

50

100

200

0 100 200 300 400 500 600 700 800 900 1000-5

10

-410

-310

-210

-110

010

110

linear wave in 1+1 dimensions, FEM

time

max

erro

r

25

50

100

200

FD

FEM SIPDG

periodic b.c., domain [0,1]

gauge condition

0 100 200 300 400 500 600 700 800 900 1000-4

10

-310

-210

linear wave in 1+1 dimensions

time

max

div

erge

nce

FD

FDM

FEM

SIPDG

NIPDG

spatial error

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

linear wave in 1+1 dimensions

x

u12

FD

FDM

FEM

SIPDG

NIPDG

exact

Traveling wave 2+1

0 100 200 300 400 500 600 700 800 900 1000-4

10

-310

-210

-110

010

110

linear wave in 2+1 dimensions, FEM

time

max

erro

r

25

50

100

200

0 100 200 300 400 500 600 700 800 900 1000-3

10

-210

-110

010

110

linear wave in 2+1 dimensions

time

max

erro

r

FD

FDM

FEM

SIPDG

periodic b.c., domain [0,1]2

quasi-uniform grid

0 100 200 300 400 500 600 700 800 900 1000-4

10

-310

-210

-110

010

110

linear wave in 2+1 dimensions, FEM

time

max

erro

r

quad

tri1

tri1*

tri2

tri2*

tri3

tri3*

Gowdy universe

0.0 0.2 0.4 0.6 0.8 1.0 1.2-6

10

-510

-410

-310

-210

-110

010

expanding Gowdy universe

time

max

erro

r

FDM

FEM

SIPDG

NIPDG

polarized gravitational waveexpanding universestart x0=1periodic b.c., domain[0,1]n=25,50,100,200

Gowdy universe

Conclusion

• Space-time FEM, DG for 2nd order waveeq.

• Unstructured space-time mesh criteria• weak formulation, FEM, DG for gravity

(Einstein-Hilbert action)