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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)