Partial Differential Equations - Modelling and ... - ResearchGate

Discontinuous Galerkin Methods 11
3 DG Approximation of an Elliptic Problem
Let Ω be a polygon in dimension d = 2 or a Lipschitz polyhedron in dimension
d = 3, with boundary ∂Ω partitioned into two disjoint parts: ∂Ω = Γ D ∪ Γ N ,
with polygonal boundaries if d = 3. For simplicity, we assume that |Γ D | is
positive. Consider the continuity equation for Darcy flow in pressure form
in Ω:
− div(K∇p) =f, in Ω, (24)
p = g 1 , on Γ D , (25)
K∇p · n Ω = g 2 , on Γ N , (26)
where n Ω is the unit normal vector to ∂Ω, exterior to Ω, and the permeability
K is a uniformly bounded, positive definite symmetric tensor, that is allowed
to vary in space. For f ∈ L 2 (Ω), g 1 ∈ H 1/2 (Γ D )andg 2 ∈ L 2 (Γ N ), system
(24)–(26) has a unique solution p ∈ H 1 (Ω) and we assume that p is sufficiently
regular to guarantee the consistency of the schemes below.
Let E h be a regular family of triangulations of Ω consisting of triangles (or
tetrahedra if d =3)E of maximum diameter h, and such that no face or side
of ∂E intersects both Γ D and Γ N . It is regular in the sense of Ciarlet [Cia91]:
There exists a constant γ>0, independent of h, such that
∀E ∈E h ,
h E
ϱ E
= γ E ≤ γ, (27)
where h E denotes the diameter of E (bounded above by h) andϱ E denotes
the diameter of the ball inscribed in E.
To simplify the discussion, we assume that E h is conforming, but most
results in this section remain valid for non-conforming grids as well as for
quadrilateral (or hexahedral if d = 3) grids. We denote by Γ h the set of all
interior edges (or faces if d =3)ofE h and by Γ h,D (resp. Γ h,N ) the set of
all edges or faces of E h that lie on Γ D (resp. Γ N ). The elements E of E h are
numbered and denoted by E i ,sayfor1≤ i ≤ P h . With any edge or face e
of Γ h shared by E i and E j with i