André Massing

A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem

We present a novel finite element method for the Stokes problem on fictitious domains. We prove inf-sup stability, optimal order convergence and uniform boundedness of the condition number of the discrete system. The finite element formulation is based on a stabilized Nitsche method with ghost penalties for the velocity and pressure to obtain stability in the presence of small cut elements. We demonstrate for the first time the applicability of the Nitsche fictitious domain method to three-dimensional Stokes problems. We further discuss a general, flexible and freely available implementation of the method and present numerical examples supporting the theoretical results.

Efficient Implementation of Finite Element Methods on Nonmatching and Overlapping Meshes in Three Dimensions

In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on nonmatching or overlapping meshes. Examples of such methods are the fictitious domain method, the extended finite element method, and Nitsches method. In all these methods, integrals must be computed over cut cells or subsimplices, which is challenging to implement, especially in three space dimensions. In this note, we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices. As a canonical example of an overlapping mesh method, we consider Nitsches method, which we apply to Poissons equation and a linear elastic problem.

A stabilized Nitsche overlapping mesh method for the Stokes problem

We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By extending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results.

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 R 3 gradient on the ...

Finite element approximation of the Laplace–Beltrami operator on a surface with boundary

We develop a finite element method for the Laplace–Beltrami operator on a surface with boundary and nonhomogeneous Dirichlet boundary conditions. The method is based on a triangulation of the surface and the boundary conditions are enforced weakly using Nitsche’s method. We prove optimal order a priori error estimates for piecewise continuous polynomials of order

A Stabilized Cut Finite Element Method for the Three Field Stokes Problem

k \ge 1

A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems

k≥1 in the energy and