consider the following model initial-value problem:
Suppose that we want to compute an approximation to u on the interval (0, T)
by using a DG method. h
u
first find a partition of the interval (0,T) and
set for n = 0, . . . , N −1. Then we look for a function which, on the interval , is the polynomial of degree at most determined by requiring that
u
hI
nk
n{ } t
n Nn=0( , 1)
n n n
I = t t +
for all polynomials v of degree at most
k
n(2)
(1)
To complete the definition of the DG method, we still need to define the quantity .
Since for the ODE,the information travels “from the past into the future”, it is reasonable to take as follows:
u
huh
This completes the definition of the DG method.
(3)
In this simple example, we already see the main components of the method, (i) The use of discontinuous approximations ,
(ii) The enforcing of the ODE on each interval by means of a Galerkin weak formulation, and
(iii) The introduction and suitable definition of the so-called numerical trace
u
hu
hThe simple choice we have made is quite natural for this case and gives rise to a very good method; however, other choices can also produce
excellent results. Next, we address the question of how to choose the
numerical trace
Let us begin with the problem of the consistency of the DG method.
As it is typical for most finite element methods, the method is said to be
consistent if we can replace the approximate solution by the exact solution u in the weak formulation (2).
We can immediately see that this is true if and only if = u.
u
uh
Next, let us consider the more subtle issue of the stability of the method.
Our strategy is to begin by obtaining a stability property for the ODE (1) which we will then try to enforce for the DG method (2) by a suitable definition of the numerical trace . uh
If we multiply the ODE(1) by u and integrate over (0, T), we get the equality
set v = uh in the weak formulation (2), integrate by parts and add over n. We get
Note that the choice corresponds to the numerical trace we chose at the beginning, namely,
Moreover, in our search for stability, we found, in a very natural way,
that the numerical trace can only depend on both traces of at t, that is, on and on .
n 1/ 2 C =
h
( )
u t u
hh
( )
u t
−u t
h( )
+Next, we want to emphasize three important properties of the DG methods that do carry over to the multi-dimensional case and to all types of problems.
The first is that the approximate solution of the DG methods does not have to satisfy any interelement continuity
constraint.
As a consequence, the method can be highly parallelizable
(when dealing with time-dependent hyperbolic problems) .
Reference:Introduction to (dis)continuous Galerkin finite element methods
by Onno Bokhove and Jaap J.W. van der Vegt
The second is that the DG methods are locally conservative.
This is a reflection of the fact that the method enforces the equation element- by-element and of the use of the numerical trace. In our simple setting, this property reads
and is obtained by simply taking v ≣ 1 in the weak formulation (2). This a much valued property in computational fluid dynamics.
The third property is the strong relation between the residuals of uh inside the intervals and its jumps across inter-interval boundaries.
To uncover it, let us integrate by parts in (2) to get
u
hNote that now we have two numerical traces, namely, and , that remain to be defined.
q
hTo do that, we begin by finding a stability result for the solution of the original equation. To do that, we multiply the first equation by q and integrate over Ω to get
Then, we multiply the second equation by u and integrate over Ω to obtain
Adding these two equations, we get
This is the result we sought. Next, we mimic this procedure for the DG method.
{ } 11 [[ ]]
{ } 12 [[ ]] 22 [[ ]]
11
h h
h h h
h h
q C u C
u C u C q
q C u n
= − × +
= − − × Ω
= − × ×
i i
h h
h
It is enough to take q 12 [[q ]] ;
u inside the domain and on its boundary
q ; u 0;
{ } 1 ( )
2
{ } 1 ( )
2 [[ ]]
[[ ]]
h
h h h
h h
where q q q
u u u
q q n q n
u u n u n
+ −
+ −
+ + − −
+ −
+ −
=
= × +
= × +
= +
= +
i i
{ } 11 [[ ]] { } 12 [[ ]] 22 [[ ]];
11 11 ( , )
h h h h h
h h
q C u C u C u C q
q C u n C n g x y
Δ Ω ∂Ω
= − × + = − − ×
= − × × + × × =
i i
h h
h
when solving - u=f in and u=g(x,y) in
ｗe take q 12 [[q ]] ; u
and on its boundary
q g(x,y) ; u ;
{ } 1 ( )
2
{ } 1 ( )
2 [[ ]]
[[ ]]
h
h h h
h h
where q q q
u u u
q q n q n
u u n u n
+ −
+ −
+ + − −
+ −
+ −
= × +
= × +
= +
= +
i i
Some properties.
(i) Let us show that to guarantee the existence and uniqueness of the approximate solution of the DG methods, the parameter C11 has to be greater than zero and the local spaces U(K) and Q(K) must satisfy the following compatibility condition:
Indeed, the approximate solution is well defined if and only if, the only approximate solution to the problem with f = 0 is the trivial solution.
In that case, our stability identity (page16) gives
which implies that qh = 0, [[uh]] = 0 on Eih, and uh = 0 on ∂Ω, provided that C11 > 0.We can now rewrite the first equation defining the method as follows:
(ii) When all the local spaces contain the polynomials of degree k, the orders of convergence of the L2-norms of the errors in q and u are k and k + 1, respectively.when C11 is of order O(h^ −1).
Example:
Domain: [-1,1]*[-1,1]
c11=1/h,c12=(0,0),c22=0,
u=x^2+y^2+x^2*y+x*y^2+x^2*y^2+x+y+x*y+1;
n=2 (n:degree of legendre polynomial)
nit u_x error(2-norm) u_y error(2-norm) u error(2-norm) 1 1.798717e-015 2.302556e-015 9.280619e-016 2 4.484038e-015 4.137567e-015 1.314049e-015 4 5.829104e-015 5.534534e-015 1.601673e-015 8 9.283783e-015 8.986503e-015 1.986222e-015 16 1.280688e-014 1.247376e-014 1.699789e-015 32 2.081884e-014 2.007377e-014 1.988631e-015
u=x^2+y^2+x^2*y+x*y^2+x^2*y^2+x+y+x*y+1;
n=1;
c11=1;c12=(0,0);c22=1;
nit u error(2-norm) 1 1.324597e+000 2 3.251126e-001 4 7.934654e-002 8 1.969471e-002 16 4.907614e-003 32 1.224563e-003
nit u Order 1 2.0158 2 2.0158 3 2.0158 4 2.0158 5 2.0158
(iii)DG methods are in fact mixed finite element methods. To see this, let us begin by noting that the DG approximate solution (qh, uh) can be
also be characterized as the solution of
which is typical of stabilized mixed finite element methods.
those methods are not well defined unless the ‘stabilizing’ form c(·, ·), usually associated with residuals, is introduced.
For DG methods, the ‘stabilizing’ form c(·, ·) solely depends on the parameter C11 and the jumps across elements of the functions in Uh.
This is why we could think that this form stabilizes the method by
penalizing the jumps, C11 being the penalization parameter;
(iv) The methods we have presented are locally conservative.
As in the hyperbolic case, this is a reflection of the form of the weak
formulation and the fact that the definition of the numerical traces on the face e does not depend on what side of it we are.
More general DG methods define the approximate solution by requiring that
for all (r, v) ∈ Q(K) × U(K).
In this general formulation, the numerical traces uh,K and qh,K can have definitions that might depend on what side of the element boundaries we are.
Hence they are not locally conservative. This is the case for the numerical fluxes in u of the last four schemes in Table 2.
the function α^r([uh]) is a special stabilization term introduced by Bassi and Rebay [12] and later studied by Brezzi et al. [20]; its stabilization properties are equivalent to the one originally presented.
λ
λ
Δ Ω ∂Ω
Ω d
Consider the Laplace eigenproblem - u= ｕ in and u=0 in where is a bounded polyhedral domain in ,d=2,3 Solving the eigenproblem with LDG method
is the eigenvalue of the matrix C-BA (-B )-1 T
