Andre Massing

Andre Massing, Benedikt Schott, Wolfgang A. Wall

We propose a stabilized Nitsche-based cut finite element formulation for the Oseen problem in which the boundary of the domain is allowed to cut through the elements of an easy-to-generate background mesh. Our formulation is based on the continuous interior penalty (CIP) method of Burman et al. [1] which penalizes jumps of velocity and pressure gradients over inter-element faces to counteract instabilities arising for high local Reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure. Since the mesh does not fit the boundary, Dirichlet boundary conditions are imposed weakly by a stabilized Nitsche-type approach. The addition of CIP-like ghost-penalties in the boundary zone allows to prove that our method is inf-sup stable and to derive optimal order a priori error estimates in an energy-type norm, irrespective of how the boundary cuts the underlying mesh. All applied stabilization techniques are developed with particular emphasis on low and high Reynolds numbers. Two- and three-dimensional numerical examples corroborate the theoretical findings. Finally, the proposed method is applied to solve the transient incompressible Navier-Stokes equations on a complex geometry.

Andre Massing, Mats G. Larson, Anders Logg et al.

We develop a Nitsche fictitious domain method for the Stokes problem starting from a stabilized Galerkin finite element method with low order elements for both the velocity and the pressure. By introducing additional penalty terms for the jumps in the normal velocity and pressure gradients in the vicinity of the boundary, we show that the method is inf-sup stable. As a consequence, optimal order a priori error estimates are established. Moreover, the condition number of the resulting stiffness matrix is shown to be bounded independently of the location of the boundary. We discuss a general, flexible and freely available implementation of the method in three spatial dimensions and present numerical examples supporting the theoretical results.

Erik Burman, Peter Hansbo, Mats G. Larson et al.

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous piecewise polynomials on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in $\mathbb{R}^3$.

Peter Hansbo, Mats G. Larson, Andre Massing

We develop a cut finite element method for the Darcy problem on surfaces. The cut finite element method is based on embedding the surface in a three dimensional finite element mesh and using finite element spaces defined on the three dimensional mesh as trial and test functions. Since we consider a partial differential equation on a surface, the resulting discrete weak problem might be severely ill conditioned. We propose a full gradient and a normal gradient based stabilization computed on the background mesh to render the proposed formulation stable and well conditioned irrespective of the surface positioning within the mesh. Our formulation extends and simplifies the Masud-Hughes stabilized primal mixed formulation of the Darcy surface problem proposed in [28] on fitted triangulated surfaces. The tangential condition on the velocity and the pressure gradient is enforced only weakly, avoiding the need for any tangential projection. The presented numerical analysis accounts for different polynomial orders for the velocity, pressure, and geometry approximation which are corroborated by numerical experiments. In particular, we demonstrate both theoretically and through numerical results that the normal gradient stabilized variant results in a high order scheme.

Erik Burman, Peter Hansbo, Mats G. Larson et al.

We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition $h k < C$, where $h$ denotes the mesh size, $k$ the wave number and $C$ a constant depending mainly on the surface curvature $κ$, but not on the surface/mesh intersection. Optimal error estimates in the $H^1$ and $L^2$-norms follow.

Andre Massing

We develop a cut Discontinuous Galerkin method (cutDGM) for a diffusion-reaction equation in a bulk domain which is coupled to a corresponding equation on the boundary of the bulk domain. The bulk domain is embedded into a structured, unfitted background mesh. By adding certain stabilization terms to the discrete variational formulation of the coupled bulk-surface problem, the resulting cutDGM is provably stable and exhibits optimal convergence properties as demon- strated by numerical experiments. We also show both theoretically and numerically that the system matrix is well-conditioned, irrespective of the relative position of the bulk domain in the background mesh.

Erik Burman, Peter Hansbo, Mats G. Larson et al.

We develop a discontinuous cut finite element method (CutFEM) for the Laplace-Beltrami operator on a hypersurface embedded in $\mathbb{R}^d$. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in $\mathbb{R}^d$. The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighboring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.

Andre Massing, Mats G. Larson, Anders Logg et al.

We present a new composite mesh finite element method for fluid--structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh which is embedded into a fixed background fluid mesh. The embedding allows for an arbitrary overlap of the fluid meshes. The coupling between the embedded and background fluid meshes is enforced using a stabilized Nitsche formulation which allows us to establish stability and optimal order \emph{a priori} error estimates, see~\cite{MassingLarsonLoggEtAl2013}. We consider here a steady state fluid--structure interaction problem where a hyperelastic structure interacts with a viscous fluid modeled by the Stokes equations. We evaluate an iterative solution procedure based on splitting and present three-dimensional numerical examples.