A-posteriori reduced basis error-estimates for a semi-discrete in space quasilinear parabolic PDE

Fabian Hoppe,Ira Neitzel

doi:10.1007/s10589-021-00299-y

Fabian Hoppe, Ira Neitzel

Open Access

https://doi.org/10.1007/s10589-021-00299-y

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Abstract

We prove a-posteriori error-estimates for reduced-order modeling of quasilinear parabolic PDEs with non-monotone nonlinearity. We consider the solution of a semi-discrete in space equation as reference, and therefore incorporate reduced basis-, empirical interpolation-, and time-discretization-errors in our consideration. Numerical experiments illustrate our results.

Highlights

In the present paper we are concerned with a-posteriori error estimation for model order reduction applied to a semi-discrete in space quasilinear parabolic partial differential equation (PDE) with non-monotone nonlinearity
In case that hyperreduction of the nonlinearity is done by empirical interpolation (EIM) we provide estimates that include the additional empirical interpolation method (EIM)-error in Sect
In order to allow for an efficient offline-online splitting, the evaluation of nonlinearities in the reduced-order model for (Eq) needs to be done by methods of hyperreduction, e.g. the Empirical Interpolation Method (EIM, [8])

Summary

Introduction

In the present paper we are concerned with a-posteriori error estimation for model order reduction applied to a semi-discrete in space quasilinear parabolic partial differential equation (PDE) with non-monotone nonlinearity. We keep the summary of results concerning (Eq) on the continuous level rather short: We refer for instance to [9, 36, 50] for a discussion of Assumption 2.1 and only recall two regularity results from the literature that might be seen as motivation for exploiting the respective regularity of the semi-discrete in space solution lateron: Regularity in case of right hand sides in the slightly more regular Bessel potentialspaces H− ,p and appropriate initial regularity has been addressed in [9, section 3.2]. Applying their setting to (Eq) yields C0, (I, W1,∞)-regularity of the solutions with some α > 0 , cf. [36, Corollary 5.4]

Semi‐discretization in space and RB‐MOR

Outlook towards optimal control

A‐posteriori RB‐error‐estimates

Some preliminary calculations

A‐posteriori estimates—Approach I

A‐posteriori estimates—Approach II

A‐posteriori RB‐ and EIM‐error‐estimates

Empirical interpolation of A

A‐posteriori RB‐error‐estimates including the EIM‐error

Numerical illustration for POD‐MOR

Test problems and technical details

Discussion of the results