Estimating spatial and parameter error in parameterized nonlinear reaction–diffusion equations

Abstract

SUMMARYA new approach is proposed for the a posteriori error estimation of both global spatial and parameter error in parameterized nonlinear reaction-diffusion problems. The technique is based on linear equations relating the linearized spatial and parameter error to the weak residual. Computable local element error indicators are derived for local contributions to the global spatial and parameter error, along with corresponding global error indicators. The effectiveness of the error indicators is demonstrated using model problems for the case of regular points and simple turning points. In addition, a new turning point predictor and adaptive algorithm for accurately computing turning points are introduced.

A patch/subdomain iteration framework and algorithm class for error indicator and mesh reliability calculations

“…In the same spirit, the approach can be applied in an artificial transient formulation to time step to a steady state and the relation to the iterative approach is again applicable [43]. In the case of nonlinear elliptic boundary-value problems, including those involving parameter continuation [44,45], nonlinear local BVP strategies may be implemented and the same framework ideas appear useful; hence this warrants further study. Analogous constructions to those advocated here can be applied for goal-oriented adaptivity of the grid.…”

Multiscale and hysteresis effects in vortex pattern simulations for Ginzburg–Landau problems

“…Note that we also experimented with pseudo-arclength continuation [29,31,32] in , but this offered little improvement for our Ginzburg-Landau problem compared to the simpler algorithm proposed above. Hence we do not consider the pseudo-arclength variant below.…”

“…We elaborate on these aspects later for specific case studies. Algorithms with continuation procedures in the parameters and w are developed next to accommodate these difficulties [29,30]. More specifically, we propose a continuation algorithm below.…”

“…Moreover, the normal flux jump across element edges (or equivalently here the jump in the gradient) corresponds to the main residual contribution from the leading scalar Laplacian operator in the governing equation. Other indicators are clearly applicable, including physicsindependent gradient recovery indicators and averaging or extrapolation techniques [35][36][37] or more 885 complete residual-based indicators [21,29], but this study is confined to gradient-jump indicators as feature indicators.…”

PID adaptive control of incremental and arclength continuation in nonlinear applications

“…This has led numerical analysts to devise special algorithms for treating the singularity such as regularizing the problem by introducing a new parameter such as an abstract arclength [4][5][6] that restores Jacobian rank at the cost of solving a bordered system [7,8]. Other special algorithms have been devised to improve performance at and near the singular point [7][8][9][10]. There are also algorithms that do not solve the bordered system but instead use this form to construct a pseudo-arclength algorithm that also enables monotone step size adjustment.…”