A note on the penalty parameter in Nitsche's method for unfitted boundary value problems

Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. From the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors

Full Gradient Stabilized Cut Finite Element Methods for Surface Partial Differential Equations

We propose and analyze a new stabilized cut finite element method for the Laplace-Beltrami operator on a closed surface. The new stabilization term provides control of the full $\mathbb{R}^3$ gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace-Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and $L^2$ norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.Comment: 20 pages, 4 figures, 5 tables. arXiv admin note: text overlap with arXiv:1507.0583

Deglutition impairment during dual task in Parkinson disease is associated with cognitive status

Introduction Dysphagia is a relevant symptom in Parkinson disease (PD), and its pathophysiology is poorly understood. To date, researchers have not investigated the effects of combined motor tasks on swallowing. Such an assessment is of particular interest in PD, in which patients have specific difficulties while performing two movements simultaneously. Objective The present study tested the hypothesis that performing concurrent tasks could decrease the safety of swallowing in PD patients as visualized using fiberoptic endoscopic evaluation of swallowing (FEES). Methods A total of 19 patients and 19 controls matched by age, gender, and level of schooling were compared by FEES under two conditions: isolated swallowing and dual task (swallowing during non-sequential opposition of the thumb against the other fingers). The two tasks involved volumes of food of 3 mL and 5 mL. The PD subjects were classified according to the Hoehn & Yahr (H&Y) Scale, the Mini Mental State Examination (MMSE), and the Montreal Cognitive Assessment (MoCA). The FEES assessment was performed according to the Boston Residue and Clearance Scale (BRACS). Results The data showed a significant worsening of swallowing in the dual task assessment for both volumes (3 mL: p ≤ 0.001; 5 mL: p ≤ 0.001) in the PD group. A correlation between the MoCA and dual-task swallowing of 3 mL was also found. Conclusion These findings suggest that additional tasks involving manual motor movements result in swallowing impairment in patients with PD. Moreover, these data highlight the need to further evaluate such conditions during treatment and assessment of PD patients

Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains

This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld and Olshanskii (ESAIM: M2AN 53(2):585–614, 2019), where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche’s method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal L2(L2) -norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples