A posteriori estimates distinguishing the error components and adaptive stopping criteria for numerical approximations of parabolic variational inequalities

Abstract

We consider in this paper a model parabolic variational inequality. This problem is discretized with conforming Lagrange finite elements of order p ≥ 1 in space and with the backward Euler scheme in time. The nonlinearity coming from the complementarity constraints is treated with any semismooth Newton algorithm and we take into account in our analysis an arbitrary iterative algebraic solver. In the case p = 1, when the system of nonlinear algebraic equations is solved exactly, we derive an a posteriori error estimate on both the energy error norm and a norm approximating the time derivative error. When p ≥ 1, we provide a fully computable and guaranteed a posteriori estimate in the energy error norm which is valid at each step of the linearization and algebraic solvers. Our estimate, based on equilibrated flux reconstructions, also distinguishes the discretization, linearization, and algebraic error components. We build an adaptive inexact semismooth Newton algorithm based on stopping the iterations of both solvers when the estimators of the corresponding error components do not affect significantly the overall estimate. Numerical experiments are performed with the semismooth Newton-min algorithm and the semismooth Newton-Fischer-Burmeister algorithm in combination with the GMRES iterative algebraic solver to illustrate the strengths of our approach.

Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation

“…Finally, Appendix C presents the application of our mass-conservative total flux reconstruction in H(div, Ω) to a challenging two-phase porous media flow problem with a finite volume fully implict/iterative coupling discretization. Applications to other problems, namely when deriving guaranteed upper bounds on the total error in presence of inexact solvers, have already been considered in [34,35] to steady and unsteady variational inequalities, in [26] to eigenvalue problems, in [60] to goal-oriented error estimates, and in [87,2] to degenerate multiphase (multicompositional) flows.…”

Adaptive Inexact Semismooth Newton Methods for the Contact Problem Between Two Membranes

“…Moreover, in contrast to these standard methods, the adaptive inexact method presented here provides an accurate estimation of the error between the exact solution and its approximation. The extension of our developments to parabolic variational inequalities is addressed in [19].…”

“…Note that (16a) corresponds to the use of a mass lumping, so that (19) for p = 1 is a local postprocess, whereas for p ≥ 2, the mass matrices in (19) are not diagonal. Extending [4,Proposition 12] to the case p ≥ 2, we can easily obtain:…”

“…Note that for p = 1, the dual basis is just the Lagrange basis ψ h,a , a ∈ V h , with the scaling 3/|ω a h |, whereas for p ≥ 2, each Θ h,x l can be determined by inverting the finite element mass matrix as in (19). An important property is that Θ h,x l belong to Λ p h for all…”

Study of weak solutions for a p(x)-Kirchhoff type variation-inequality problems arising from financial problems

“…In recent years, the initial boundary value problems based on variational inequalities have gradually attracted the attention of scholars [10][11][12][13]. Compared with the traditional parabolic initial boundary value problem, variationinequality one adds one or more inequality constraints based on parabolic operator and initial boundary value.…”