Finite element approximation to linear, second order, parabolic problems with $L^1$ data

TL;DR

This work addresses the finite element approximation of the heat equation with $L^1$ data by employing a mass-lumped implicit Euler scheme in time and linear finite elements in space, within the framework of renormalized solutions. The authors establish convergence to the renormalized solution under an inverse CFL condition and a discrete maximum principle on the mesh, along with a conditional inf-sup stability result for the scheme. Key contributions include a priori estimates based on truncations that control the solution in $L^ fty(0,T;L^1(\\Omega))$ and in weak-$L^{\\overline{q}}$ spaces for the spatial gradient, and a convergence result in $L^ abla(0,T;W^{1,q}_0)$ for all $q<\\overline{q}=(d+2)/(d+1)$. The results provide a rigorous foundation for numerically solving parabolic problems with very low-regularity data and suggest avenues for extending to variable coefficients, reaction-diffusion terms, and more challenging convection-dominated regimes.

Abstract

We consider the approximation to the solution of the initial boundary value problem for the heat equation with right hand side and initial condition that merely belong to $L^1$. Due to the low integrability of the data, to guarantee well-posedness, we must understand solutions in the renormalized sense. We prove that, under an inverse CFL condition, the solution of the standard implicit Euler scheme with mass lumping converges, in $L^\infty(0,T;L^1(Ω))$ and $L^q(0,T;W^{1,q}_0(Ω))$ ($q<\tfrac{d+2}{d+1}$), to the renormalized solution of the problem.

Theorems & Definitions (17)

• Definition 2.1: renormalized solution
• Theorem 2.2: renormalized solutions
• Proposition 3.2: mass lumping
• Proof 1
• Lemma 3.3: nonlinear estimate
• Proof 2
• Lemma 3.4: truncation vs. interpolation
• Proof 3
• Theorem 3.5: truncations
• Proof 4