Abstract
We consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an hp-version discontinuous Galerkin (DG) time stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds resulting, for instance, in concrete bounds in - and -type norms when I is the temporal and the spatial domain for the PDE. We base our methodology for the analysis on a novel space-time reconstruction approach. Our approach is flexible as it works for any type of elliptic error estimator and leaves their choice to the user. It also exhibits mesh-change estimators in a clear and concise way. We also show how our approach allows the derivation of such bounds in the norm.
Highlights
Adaptive numerical methods have been shown to provide accurate and efficient numerical treatment of evolution partial differential equations (PDEs) thanks to their properties for localized mesh resolution especially in the context of moving fronts, interfaces, singularities, or layers
Some of the classical works on adaptive finite element methods for parabolic problems [10,11,12,13,14] are based on discontinuous Galerkin (DG) time stepping combined with FEM in space, and proving a posteriori bounds in various norms using duality techniques
We present a posteriori error bounds for an hp-version DG-in-time and conforming Galerkin discretization in space method for both L∞(I ; L2)- and L2(I ; H1)-norm errors separately, allowing for what appears to be optimal order in each case
Summary
Adaptive numerical methods have been shown to provide accurate and efficient numerical treatment of evolution PDEs thanks to their properties for localized mesh resolution especially in the context of moving fronts, interfaces, singularities, or layers (both boundary and interior). More recent results on rigorous a posteriori bounds for parabolic problems have focused on extending the paradigm of the reliable and efficient a posteriori error analysis of elliptic problems to the parabolic case [31,36,37] Such works typically involve basic low-order time stepping schemes combined with various types of FEM in space. To facilitate a wide range of applications, we will present the theory within a Gelfand-triple-type abstract setting allowing, for instance, both second- and fourth-order spatial operators This generality comes at the possible expense of different, yet quantitatively analogous, computable constants in the resulting a posteriori error estimators compared to the bounds in [16,17,19].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
