2018/03/08 by Oliver J. Sutton, Sutton, Oliver J. · 3 citations
Engineering · Physics and Astronomy · #65M15 #65M60 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1803.03207
openalex publication_date 2018/03/08 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Computable estimates for the error of finite element discretisations of\nparabolic problems in the L^\∞(0,T; L2) norm are developed, which\nexhibit constant effectivities (the ratio of the estimated error to the true\nerror) with respect to the simulation time. These estimates, which are of\noptimal order, represent a significant advantage for long-time simulations, and\nare derived using energy techniques based on elliptic reconstructions. The\neffectivities of previous optimal order error estimates in this norm derived\nusing energy techniques are shown numerically to grow either in proportion to\nthe simulation duration or its square root, a key disadvantage compared with\nearlier estimators derived using parabolic duality arguments. The new estimates\nform a continuous family, almost all of which are new, reproducing certain\nfamiliar energy-based estimates well suited for short-time simulations and not\navailable through the parabolic duality framework. For clarity, we demonstrate\nthe technique applied to a linear parabolic problem discretised using standard\nconforming finite element methods in space coupled with backward Euler and\nCrank-Nicolson time discretisations, although it can be applied much more\nwidely.\n