2012/12/23 by Erwan Faou, Alexander Ostermann, Faou, Erwan +3 · 1 citation
Computer Science · Mathematics · #65J08 #65L12 #65M15 #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1212.5827
openalex publication_date 2012/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We analyze the convergence of the exponential Lie and exponential Strang splitting applied to inhomogeneous second-order parabolic equations with Dirichlet boundary conditions. A recent result on the smoothing properties of these methods allows us to prove sharp convergence results in the case of homogeneous Dirichlet boundary conditions. When no source term is present and natural regularity assumptions are imposed on the initial value, we show full-order convergence of both methods. For inhomogeneous equations, we prove full-order convergence for the exponential Lie splitting, whereas order reduction to 1.25 for the exponential Strang splitting. Furthermore, we give sufficient conditions on the inhomogeneity for full-order convergence of both methods. Moreover our theoretical convergence results explain the severe order reduction to 0.25 of splitting methods applied to problems involving inhomogeneous Dirichlet boundary conditions. We include numerical experiments to underline the sharpness of our theoretical convergence results.