DG for steady diffusion equations

The goal of this section is to review the IP-DG semidiscretization of the steady homogeneous and heterogeneous diffusion equations define as follow







−∇ · (∇u) = f on Ω

u = 0 on ∂Ω

, (5.31)

where f ∈ L²(Ω) and  ∈ R⁺ are respectively the source term and the diffusion coefficient. Let us remind that we have used IP-DG method, introduced in by Wheeler [220] and Arnold [11], to investigate the one dimensional diffusion equation in Chapter 3. Here, we consider (5.31) in a dimension d ≥ 2

Let us assume that the diffusion coefficient  is constant on Ω (i.e. homogeneous).

From the weak form of (5.31) at the continuous level, we design the IP-DG method

is design to approximate the solution of (5.31) in L²−setting. In order to derive the weak form of (5.31) at the continuous level, the solution u is assumed to belong to the space V = H₀¹(Ω) = {v ∈ H¹(Ω) such that v |_∂Ω= 0}. Then by the means of the integration by parts, the weak form of (5.31) is defined as follows

Find u ∈ V s.t. b(u, w) = Z

Ω

f w, ∀w ∈ V, (5.32)

where the bilinear form b is given by

b(u, w) = Z

Ω

∇u · ∇w. (5.33)

Note that the bilinear form b(·, ·) is symmetric and bounded in V × V . According to the Poincar´e inequality, see [101], there is a constant C_Ω such that for all v ∈ V

| b(v, v) |≥  k ∇v k_[L²_(Ω)]^d≥ 

C_Ω k v k_L²_(Ω) . (5.34)

Then, owing to Theorem 2.1, the weak form (5.32) is well-posed.

Now that the weak form of (5.31) at the continuous level is reviewed, we now focus on the design of the IP-DG method to approximate the solution of (5.32) by a solution in V_h. To this end, we mimic at the discrete level the properties of the bilinear form b(·, ·) (i.e. symmetry, L²−coercivity and boundness) that holds at the continuous level, by the means of some penalty terms. Therefore the integration by parts is operated on each element T of T . This applied to (5.31), leads to

B_d(u, w_h) = Z

Ω

f w_h, (5.35)

where B_d(·, ·) is the bilinear function defined as follows

B_d(u, w_h) = X transform the second term in the right hand side of (5.36) into a sum of integrals

over the the faces as follows

As for the 1D case in Chapter 3, by using (3.27) with the definition of the jump and the average functions in (5.25)-(5.29), we obtain

[[∇_huw_h]] =

Since  is constant, then according to [80], by assuming that the exact solution u ∈ V ∩ W^2,1(Ω), we have

∀F ∈ F_h, [[u]] = 0 and ∀F ∈ F_hⁱ, [[∇u]] = 0. (5.39)

Therefore, combining (5.38) and (5.39) in (5.37) yields

X

and then the bilinear form B_d(·, ·) becomes

Bd(u, wh) = X

Note from (5.40) that the bilinear form B_d(·, ·) is nonsymmetric, owing to the second term on the right hand side of (5.40). Since a desirable property of the discrete bilinear form is to preserve the original symmetry of the bilinear form b(·, ·), we consider the bilinear form for all v, w_h ∈ V_h. Indeed, symmetry can simplify the resolution process of the resulting linear system and furthermore, it is a natural ingredient to derive optimal

L²−norm error estimation [11, 80]. The second term on the right-hand side of (5.41) is called the symmetric term.

Other desirable properties are the discrete L²−coercivity and boundness on the broken polynomial space V_hwith respect to a suitable norm. Indeed, these properties will ensure the well posedness of the discrete weak form, owing to Theorem 2.1. But the difficulty with the discrete bilinear form B_d^s(·, ·) defined by (5.41) is that, for all w_h ∈ V_h

and the second term on the right-hand side has no a priori sign so that, without adding a further term, there is no hope for discrete coercivity. To achieve discrete coercivity, we add to B_d^s(·, ·), defined by (5.41), a term penalizing interface and boundary jumps. Namely we set as in [11, 80]

B_d^sip(u_h, w_h) = B_d^s(u_h, w_h) +

where h_F is a local length scale associated with the face F ∈ F_h, the quantity η > 0 denotes a user-dependent parameter which is independent of the diffusion coefficient. The local length scale hF is set to the diameter of the face F in the dimension d ≥ 2. If the penalty parameter η is large enough then the bilinear form B_d^sip(·, ·) is coercive in V_h, see [11, 80]. The bilinear form B_d^sip(·, ·) is called the SIPG bilinear form. Note from (5.42) that the discrete bilinear form B_d^sip(·, ·) is still symmetric since the bilinear forms B_d^s(·, ·) and the so-called penalty bilinear form S_h are symmetric. Due to (5.39) for all w_h ∈ V_h and the exact solution u, we have B_d^sip(u, w_h) = B_d(u, w_h). This shows that the consistency of B_d has not changed with the terms added. Therefore, owing to Theorem 2.1, a well-posed weak form of (5.31) can be formulated as follows

Find u_h ∈ V_h s.t. B_d^sip(u_h, w_h) = Z

Ω

f w_h, ∀w_h ∈ V_h. (5.43)

Now that the SIPG weak form has been presented for the steady diffusion

equa-tion subject to the nonhomogeneous Dirichlet boundary condiequa-tion, we focus on the formulation of the SIPG weak form in the case of homogeneous Dirichlet, Neumann and Robin boundary condition. The importance of this investigation is to be able to weakly enforce any type of boundary condition.

• Firstly, let us consider the steady diffusion equation subject to nonhomoge-neous Dirichlet boundary condition i.e. u = g on ∂Ω, with g ∈ L²(∂Ω). In this case the jump function of u is no longer equal to zero across the external face F ∈ F_h^e, as in (5.39). Then the weak form of (5.31) can be formulated as

for all w_h in V_h. The second and third terms on the right-hand side of (5.44) appeared while adding the terms to mimic at the discrete level the symmetry and the coercivity of the continuous weak form.

• Secondly, let us consider the steady diffusion equation subject to homogeneous Neumann boundary condition i.e. n · ∇u = 0 on ∂Ω. In this case the jump function of the diffusive flux is equal to zero across the external face F ∈ F_h^e. Then (5.40) can be reduced as follows

B_d(u_h, w_h) = X

By adding the terms to mimic at the discrete level the symmetry and the coercivity of the continuous weak form, we can formulate the weak form of (5.31) as follows : find u_h in V_h such that

A(uh, wh) = Z

Ω

f wh ∀wh ∈ Vh,

where the bilinear term A is defined by

• Finally, let us consider the steady diffusion equation subject to the Robin boundary condition λu + n · ∇u = g on ∂Ω, with g ∈ L²(∂Ω), λ ∈ L^∞(∂Ω) and λ non negative almost everywhere on ∂Ω. Then the weak form of (5.31) can be formulated as follows : find uh in Vh such that

A(u_h, w_h) + X

To derive the weak form in this case, we estimate the jump function of the diffusive flux (i.e. n · ∇u) from the boundary condition, then we substitute the result in (5.40) and follow the process to mimic at the discrete level the symmetry and the coercivity of the continuous weak form.

We examine the convergence of the IP-DG method reviewed here for an unsteady and homogeneous diffusion equation in Section 5.3.2.

Theorem 5.1 (IP-DG for heterogeneous diffusion equation). Let assume that there is a partition of Ω in a set P_Ω = {Ω_i, i = 1, · · · , n} of polyhedrons such that the restriction of  to each polyhedron Ω_i is constant. If the mesh T_h is such that each element T ∈ T_h belongs to only one polyhedron Ω_i then the weak form of (5.31) takes the form

Find u_h ∈ V_h s.t. B_d^swip(u_h, w_h) = Z

Ω

f w_h, ∀w_h ∈ V_h, (5.46) where the bilinear form B_d^swip is given by

B_d^swip(u_h, w_h) =X

Here, the weighted average {·}^F_w and the diffusion dependent penalty parameter γ_F^

The bilinear form B_d^swip, called the symmetric weighted interior penalty galerkin (SWIPG) method, was introduced in 2003 by Dryja [87] and further analysed by Ern et al. [81, 98]. Note that if the diffusion coefficient is constant on Ω, the bilinear forms B_d^swip and B_d^sip are equal. In Section 5.3.5, we use B_d^swip to simulate the velocity of the fluid trough a medium with fracture.