We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational me
thod. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.
In this paper we discuss a hybridised method for FEM-BEM coupling. The coupling from both sides use a Nitsche type approach to couple to the trace variable. This leads to a formulation that is robust and flexible with respect to approximation spaces
and can easily be combined as a building block with other hybridised methods. Energy error norm estimates and the convergence of Jacobi iterations are proved and the performance of the method is illustrated on some computational examples.
In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements. To obtain stability we add a penalty on the jump of the gradient over elem
ent faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that in the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally the $L^2$-norm error converges with the expected order $O(h^{k+frac12})$. We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if discontinuities have been present in the solution at some time during the evolution.
We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a
consistent discretization depending on a weighting parameter scaling as $mathcal O(h^{-1})$, while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as $mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual $L^2$-norm. However, we are still able to recover the optimal a priori $L^2$-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.
We present a new approach for the mechanically consistent modelling and simulation of fluid-structure interactions with contact. The fundamental idea consists of combining a relaxed contact formulation with the modelling of seepage through a porous l
ayer of co-dimension 1 during contact. For the latter, a Darcy model is considered in a thin porous layer attached to a solid boundary in the limit of infinitesimal thickness. In combination with a relaxation of the contact conditions the computational model is both mechanically consistent and simple to implement. We analyse the approach in detailed numerical studies with both thick- and thin-walled solids, within a fully Eulerian and an immersed approach for the fluid-structure interaction and using fitted and unfitted finite element discretisations.
In the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moder
ate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the $L^2$-norm for smooth flows in the pre-asymptotic high Reynolds number regime.
We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the bo
undary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More precisely, the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.
We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection--diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank-Ni
colson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the $tau^2 + h^{p+{frac12}}$ error estimates for the $L^2$-norm under either the standard hyperbolic CFL condition, when piecewise affine ($p=1$) approximation is used, or in the case of finite element approximation of order $p ge 1$, a stronger, so-called $4/3$-CFL, i.e. $tau leq C h^{4/3}$. The theory is illustrated with some numerical examples.
In this note we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the
nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples.
In this paper we discuss a level set approach for the identification of an unknown boundary in a computational domain. The problem takes the form of a Bernoulli problem where only the Dirichlet datum is known on the boundary that is to be identified,
but additional information on the Neumann condition is available on the known part of the boundary. The approach uses a classical constrained optimization problem, where a cost functional is minimized with respect to the unknown boundary, the position of which is defined implicitly by a level set function. To solve the optimization problem a steepest descent algorithm using shape derivatives is applied. In each iteration the cut finite element method is used to obtain high accuracy approximations of the pde-model constraint for a given level set configuration without re-meshing. We consider three different shape derivatives. First the classical one, derived using the continuous optimization problem (optimize then discretize). Then the functional is first discretized using the CutFEM method and the shape derivative is evaluated on the finite element functional (discretize then optimize). Finally we consider a third approach, also using a discretized functional. In this case we do not perturb the domain, but consider a so-called boundary value correction method, where a small correction to the boundary position may be included in the weak boundary condition. Using this correction the shape derivative may be obtained by perturbing a distance parameter in the discrete variational formulation. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem.
