# consider the following model initial-value problem:

## Full text

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### 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

h

n

n

### { } t

n Nn=0

( , 1)

n n n

I = t t +

for all polynomials v of degree at most

### k

n

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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

h

uh

This completes the definition of the DG method.

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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

h

h

### numerical trace

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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

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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

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n 1/ 2 C =

h

h

h

h

+

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### (when dealing with time-dependent hyperbolic problems) .

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Reference:Introduction to (dis)continuous Galerkin finite element methods

by Onno Bokhove and Jaap J.W. van der Vegt

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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.

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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

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### u

h

Note that now we have two numerical traces, namely, and , that remain to be defined.

### q

h

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To 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.

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h h

h h h

h h

h h

h

h

h h h

h h

+

+

+ +

+

+

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h h h h h

h h

h h

h

h

h h h

h h

+

+

+ +

+

+

### i i

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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

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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:

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(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]

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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

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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

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(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

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which is typical of stabilized mixed finite element methods.

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### penalizing the jumps, C11 being the penalization parameter;

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(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.

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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.

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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

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