Moving Dirichlet boundary conditions.

*(English)*Zbl 1314.65123The paper is concerned with the second-order initial value problem for the system \(\mathcal{M}\ddot{u}+\mathcal{D}\dot{u}+\mathcal{K}u=\mathcal{F}(t)\) subject to boundary conditions on a time-dependent part \(\Gamma_D(t) \subset \partial\Omega\) of the boundary of a bounded Lipschitz domain \(\Omega \subset \mathbb{R}^n\). As an example for this model an elastic body \(\Omega\) which is coupled with a spring damper system can serve.

In the presented method, the time-dependent boundary condition is taken into account as a weak constraint incorporated in the Lagrange multiplier method. A time-dependent ansatz space in the finite element method considered for discretization is avoided by working with a bi-Lipschitz transformation of the moving boundary on a \((n-1)\)-dimensional domain \(I\).

Solutions of the given system are considered to be elements of \(\mathcal{V}:=[H^1(\Omega)]^n\) and the problem is studied in weak form in the setting of the Gelfand triple \((\mathcal{V},\mathcal{H},\mathcal{V^*})\), where \(\mathcal{V}:=[L^2(\Omega)]^n\). The boundary condition is included as a weak constraint with the aid of a bilinear form \(b\). As a result a saddle-point problem is obtained. It is proved that \(b\) satisfies the inf-sup condition and consequently the existence of a Lagrange multiplier can be shown.

As finite elements continuous piecewise linear functions in the interior of a triangulation and bubble functions for the boundary edges are taken (in this part of the paper \(n=2\) is assumed). The Lagrange multiplier is approximated by piecewise constant functions on a partition of \(I\). The main result of the paper is the proof of the discrete inf-sup condition. The setting of the weak formulation including the Lipschitz transformation requires the handling of various Sobolev spaces and its adjoints which is thoroughly worked out.

In the presented method, the time-dependent boundary condition is taken into account as a weak constraint incorporated in the Lagrange multiplier method. A time-dependent ansatz space in the finite element method considered for discretization is avoided by working with a bi-Lipschitz transformation of the moving boundary on a \((n-1)\)-dimensional domain \(I\).

Solutions of the given system are considered to be elements of \(\mathcal{V}:=[H^1(\Omega)]^n\) and the problem is studied in weak form in the setting of the Gelfand triple \((\mathcal{V},\mathcal{H},\mathcal{V^*})\), where \(\mathcal{V}:=[L^2(\Omega)]^n\). The boundary condition is included as a weak constraint with the aid of a bilinear form \(b\). As a result a saddle-point problem is obtained. It is proved that \(b\) satisfies the inf-sup condition and consequently the existence of a Lagrange multiplier can be shown.

As finite elements continuous piecewise linear functions in the interior of a triangulation and bubble functions for the boundary edges are taken (in this part of the paper \(n=2\) is assumed). The Lagrange multiplier is approximated by piecewise constant functions on a partition of \(I\). The main result of the paper is the proof of the discrete inf-sup condition. The setting of the weak formulation including the Lipschitz transformation requires the handling of various Sobolev spaces and its adjoints which is thoroughly worked out.

Reviewer: Rolf Dieter Grigorieff (Berlin)

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

35L05 | Wave equation |