Abstract
We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω and the right-hand side belongs to L1Ω; we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in W01,qΩ for every q with 1≤q<d/d-1 (d=2 or d=3) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate in W01,qΩ when the right-hand side f belongs to LrΩ verifying Tkf∈H1Ω for every k>0, for some r>1.
Highlights
In this work we consider, in dimension d = 2 or 3, the P1 discontinuous Galerkin method approximation of the Dirichlet problem−div (A∇u) = f in Ω, (1)u = 0 on ∂Ω, where Ω is an open bounded set of Rd, A is a coercive matrix with coefficients in L∞(Ω), and f belongs to L1(Ω).The solution of (1) does not belong to H01(Ω) for a general right-hand side in L1(Ω)
We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞(Ω) and the right-hand side belongs to L1(Ω); we extend the results where the case of linear finite elements approximation is considered
We prove that the unique solution of the discrete problem converges in W01,q(Ω) for every q with 1 ≤ q < d/(d − 1) (d = 2 or d = 3) to the unique renormalized solution of the problem
Summary
In order to correctly define the solution of (1), one has to consider a specific framework, the concept of renormalized (or equivalently entropy) solution (see for example [1, 2]). These definitions allow one to prove that in this new sense problem (1) is wellposed in the terminology of Hadamard. (v) If N designate the number of all interior centers mi of faces F in Th we define the interpolation operator Πh and the truncated interpolation operator Ihk by ∀V ∈ L2 (Ω) avec ∫ [Vh] = 0, Πh (V) ∈ Vh, Πh (V) fl ∑ αiVφi,. ∀i ∈ {1, 2, . . . , N} : Qii − ∑ Qij ≥ 0
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