On the a posteriori error analysis for linear Fokker–Planck models in convection-dominated diffusion problems

Abstract

This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convectiondominated diffusion problems motivated by a linear Fokker-Planck problem appearing in computational neuroscience. We obtain computable error bounds of functional type for the static and time-dependent case and for different boundary conditions (mixed and pure Neumann boundary conditions). Finally, we present a set of various numerical examples including discussions on mesh adaptivity and space-time discretisation. The numerical results confirm the reliability and efficiency of the error estimates derived.numerical examples for the linear Fokker-Planck model problem for different parameters and boundary conditions. A specially interesting case is the convection-dominated (CD) setting of the problem, which may follow from various applications, such as the linearised Navier-Stokes equation with high Reynolds number or drift-diffusion equations of a semiconductor device modelling. Due to the small diffusion coefficient, the solution of the discussed problem has singularities in form of boundary or interior layers. The classical numerical approximations relying on equidistant meshes are not able to capture these layers unless the mesh contains an unacceptably large amount of nodes. If the mesh is not fine enough, the obtained approximations will include the non-physical oscillations polluting the data. The techniques developed to trigger the above described issue include the upwind scheme [22], streamline diffusion finite element method (SDFEM), also known as streamline-upwind/Petrov-Galerkin formulation (SUPG) [12], residual free bubble (RFB) functions [7,8,7,9, 10], Galerkin Least Squares (GLS) [25,2,47], Continuous Interior Penalty (CIP) [19], Edge Stabilisation (ES) [14], exponential fitting [4, 11,53,57], discontinuous Galerkin methods [6,24], and spurious oscillations at layers diminishing (SOLD) methods [27,28]. This list is by no means exhaustive, and there exist other techniques tackling this issue. In this work, we use the SUPG method introduced by Brooks and Hughes [12].An alternative approach to handle the state Fokker-Planck boundary value problem, see later (12)-(13), relies on an adaptation (grading) of the underlying meshes in order to capture the boundary layers. The mesh adaptation for convection-dominated problems (CDPs) affects both the stability and the accuracy of the scheme. In practice, it was observed that once the accuracy is improved through the mesh adaption, the scheme gets stabilised as well (see, e.g., [5,47,58]). According to the numerical results presented in [31,47,58] and the theoretical justification for a model problem discussed in [47,59], meshes can be adapted to boundary levels in such a way that even a standard finite element method can provide the optimal or quasi-optimal approximation property. However, the question of constructing such a grid, correctly adapted to the particular problem, still remains open. Moreover, in [51] the authors conclude that the stabilisation of...

Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

“…First functional type estimates for inverse problems, which are related to optimal control problems, can be found in [41,8]. Moreover, recent results on guaranteed computable estimates for convection-dominated diffusion problems are presented in [33]. In [26], majorants for one cost functional of a time-periodic parabolic optimal control and for the corresponding optimality system were presented.…”

Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

“…First functional type estimates for inverse problems, relatives to optimal control problems, can be found in [41,8]. Moreover, recent results on guaranteed computable estimates for convection-dominated diffusion problems are presented in [33]. In [26], majorants for one cost functional of a time-periodic parabolic optimal control and for the corresponding optimality system were presented.…”