Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems

Abstract

This work is concerned with the derivation of a robust a posteriori error estimator for a discontinuous Galerkin method discretisation of a linear non-stationary convection-diffusion initial/boundary value problem and with the implementation of a corresponding adaptive algorithm. More specifically, we derive a posteriori bounds for the error in the L 2 (H 1 ) + L ∞ (L 2 )-type norm for an interior penalty discontinuous Galerkin (dG) discretisation in space and a backward Euler discretisation in time. Finally, an adaptive algorithm is proposed utilising the error estimator. Optimal rate of convergence of the adaptive algorithm is observed in a number of test problems and for various Pèclet numbers.

Time-space adaptive discontinuous Galerkin method for advection-diffusion equations with non-linear reaction mechanism

“…In this work, we construct the residual-based, robust (in Péclet number) a posteriori error estimators for non-stationary convection domianted diffusionconvection equations with non-linear reaction mechanisms. To construct the a posteriori error estimators, we extend the a posteriori error estimators for linear nonstationary problems constructed in [7] which uses the a posteriori error estimators for linear stationary models constructed in [17] utilizing the elliptic reconstruction technique [14] to make connection between the stationary and non-stationary error. As in [7], first we construct and prove the a posteriori error bounds for non-linear stationary models in Section 4.1.…”

“…To construct the a posteriori error estimators, we extend the a posteriori error estimators for linear nonstationary problems constructed in [7] which uses the a posteriori error estimators for linear stationary models constructed in [17] utilizing the elliptic reconstruction technique [14] to make connection between the stationary and non-stationary error. As in [7], first we construct and prove the a posteriori error bounds for non-linear stationary models in Section 4.1. Then, in Section 4.2, we give the a posteriori error bounds for the semi-discrete system of the non-stationary problems with non-linear reaction term.…”

“…By this way, we are able to obtain results being optimal order in both L 2 (H 1 ) and L ∞ (L 2 )-type norms, while the results obtained by the standard energy methods are only optimal order in L 2 (H 1 )-type norms, but sub-optimal order in L ∞ (L 2 )-type norms. In [7], a posteriori error estimates in the L ∞ (L 2 )+L 2 (H 1 )-type norm are derived for linear parabolic diffusion-convection-reaction equations using backward Euler in time and discontinuous Galerkin in space utilizing the elliptic reconstruction technique. In this paper, we extend the study in [7] in a similar way by deriving and implementing a posteriori error estimates in the L ∞ (L 2 ) + L 2 (H 1 )type norm using backward Euler in time and discontinuous Galerkin (symmetric interior penalty (SIPG)) in space for the convection dominated parabolic problems with non-linear reaction mechanisms.…”

Virtual Element Method for the Laplace-Beltrami equation on surfaces

“…The increased mesh generality provides different advantages, we mention some of them. Nonconforming meshes (i) naturally arise when pasting several meshes to obtain a polygonal approximation of the whole domain [10,11], as there is no need to match the nodal points in contrast to conforming pasting techniques [12,13] and (ii) allow simple adaptive refinement strategies [14]. Elements of more general shape and arbitrary number of edges allow (i) flexible approximation of the domain and in particular of its boundary [15] and (ii) the possibility of enforcing higher regularity to the numerical solution [6,16,17].…”

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

“…Elliptic reconstruction technique was developed in [12,16] for parabolic problems, and extended to [3] for non-stationary convection diffusion equations, Demlow et al [5], Kopteva and Linss [11] for maximum norm a posteriori error estimates as well as [10,18] for parabolic integro-differential equation and Schrodinger equation. The elliptic reconstruction can be viewed as an a posteriori analogue to the Ritz-elliptic projection used in a priori error analysis for parabolic equations.…”