Discontinuous Galerkin method for linear parabolic equations with L^1-data

Abstract

In this work, we examine the discontinuous Galerkin method for parabolic linear problem with data in L¹(Ω × (0, T)). On one hand, using a Euler time advancing scheme that goes backwards, we can discretize a time interval. Furthermore, the discretization of space is based on Symmetric Weighted Interior Penalty (SWIPG) method. We use the technique of construction of the renormalized solution to obtain existence of the discrete solution. Then, our research demonstrates that the discrete solution converges in L¹(Q) to the unique renormalized solution of the problem, where Q= Ω × (0, T). In the case where the coefficients are smooth, we offer an estimate of the error in L¹(Q), when the side on the right is assigned to Marcinkiewicz space L^{s, ∞}(Q) where 1 < s < 2.