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Moving Dirichlet boundary conditions. (English) Zbl 1314.65123
The 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.
##### 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
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