Finite element approximation to linear, second order, parabolic problems with L¹ data

2025/10/06 by Gabriel R. Barrenechea, Barrenechea, Gabriel, Abner J. Salgado +1
Engineering · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Numerical methods in engineering #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2510.05331

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 L¹. 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^∞(0,T;L¹(Ω)) and L^q(0,T;W^1,q₀(Ω)) (q<\tfracd+2d+1), to the renormalized solution of the problem.

Citations

• Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations
• Lagrangian solutions to the Vlasov-Poisson system with L¹ density

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