## Abstract

A space-time adaptive algorithm to solve the motion of a rigid disk in an incompressible Newtonian fluid is presented, which allows collision or quasi-collision processes to be computed with high accuracy. In particular, we recover the theoretical result proven in [M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 2007, 7–9, 1345–1371], that the disk will never touch the boundary of the domain in finite time. Anisotropic, continuous piecewise linear finite elements are used for the space discretization, the Euler scheme for the time discretization. The adaptive criteria are based on a posteriori error estimates for simpler problems.

## A Derivation of the Penalty Method (2.1)

We present the derivation of the penalty formulation (2.1) used to approximate the motion of the rigid disk inside the cavity. We start from the physical model presented in [12]. We study the motion of a rigid disk of radius *R* inside a bounded, convex cavity

together with

where
*m* and *J* its mass and inertia. In the above system,

Note that since

which is also true for any constant vector

To derive equations (2.1), the first step consists to establish a variational formulation that can easily be approximated with Galerkin methods, that is to say involving test functions that are defined on the whole cavity Ω. We follow the formulation introduced in [15]. Let us consider the problem at a given time *t*, and to lighten the presentation, we drop the dependence of

that has the equivalent definition (see for instance [27])

We now multiply the momentum equation of (A.1) by a test function

one has after integration by part

Using the Newton’s laws (A.2) and the fact that

Moreover, one has that

and

Thus since

which leads to

Finally, since

where we note

The constrains

where

Written in the strong form, the approximated variational formulation gives

with

Observe that the last equation reduces to

## B Justification of (2.8)

We now prove an a posteriori error estimate for the time discretization of a simplified problem. We simplify the first equation of (2.1) and consider the following problem:

where

We consider the forward Euler method. Given a integer

then

We assume that there exists a unique

solution of (B.1),
that

Observe that problem (B.1) reads: starting from

where, for any

and

where

## Lemma 1.

*Let
*

*where
*

*where*

The key point to prove this lemma consists in mapping
*d*) is sharp.

Our main result is then the following.

## Theorem 2.

*Let
*

## Proof.

Let

Let

Taking the scalar product of the last equation with

Without loss of generality, we assume that

with

Multiplying the last inequality by

and
integrating from

Summing up over *n* leads to

where we use that

## Acknowledgements

Frédéric Alauzet is acknowledged for providing the Wolf-Interpol program corresponding to conservative interpolation [3]. Samuel Dubuis acknowledges Diane Guignard and Swarnendu Sil for advices and discussions concerning the derivation of the time error indicator. It should be noted that some numerical experiments with space adaptation only were already reported in [26].

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**Received:**2020-03-31

**Revised:**2020-07-31

**Accepted:**2020-10-23

**Published Online:**2020-11-07

**Published in Print:**2021-04-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston