A Cut Finite Element Method with Boundary Value Correction

In this contribution we develop a cut finite element method with boundary value correction of the type originally proposed by Bramble, Dupont, and Thomee. The cut finite element method is a fictitious domain method with Nitsche type enforcement of Dirichlet conditions together with stabilization of the elements at the boundary which is stable and enjoy optimal order approximation properties. A computational difficulty is, however, the geometric computations related to quadrature on the cut elements which must be accurate enough to achieve higher order approximation. With boundary value correction we may use only a piecewise linear approximation of the boundary, which is very convenient in a cut finite element method, and still obtain optimal order convergence. The boundary value correction is a modified Nitsche formulation involving a Taylor expansion in the normal direction compensating for the approximation of the boundary. Key to the analysis is a consistent stabilization term which enables us to prove stability of the method and a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary

Comparison of Shape Derivatives Using CutFEM for Ill-posed Bernoulli Free Boundary Problem

In this paper we study and compare three types of shape derivatives for free boundary identification problems. The problem takes the form of a severely ill-posed Bernoulli problem where only the Dirichlet condition is given on the free (unknown) boundary, whereas both Dirichlet and Neumann conditions are available on the fixed (known) boundary. Our framework resembles the classical shape optimization method in which a shape dependent cost functional is minimized among the set of admissible domains. The position of the domain is defined implicitly by the level set function. The steepest descent method, based on the shape derivative, is applied for the level set evolution. For the numerical computation of the gradient, we apply the Cut Finite Element Method (CutFEM), that circumvents meshing and re-meshing, without loss of accuracy in the approximations of the involving partial differential models. We consider three different shape derivatives. The first one is the classical shape derivative based on the cost functional with pde constraints defined on the continuous level. The second shape derivative is similar but using a discretized cost functional that allows for the embedding of CutFEM formulations directly in the formulation. Different from the first two methods, the third shape derivative is based on a discrete formulation where perturbations of the domain are built into the variational formulation on the unperturbed domain. This is realized by using the so-called boundary value correction method that was originally introduced to allow for high order approximations to be realized using low order approximation of the domain. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem

A cut finite element method for a model of pressure in fractured media

We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: (1) Stability of the formulation in the full range of parameter choices; and (2) Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples

Galerkin least squares finite element method for the obstacle problem

We construct a consistent multiplier free method for the finite element solution of the obstacle problem. The method is based on an augmented Lagrangian formulation in which we eliminate the multiplier by use of its definition in a discrete setting. We prove existence and uniqueness of discrete solutions and optimal order a priori error estimates for smooth exact solutions. Using a saturation assumption we also prove an a posteriori error estimate. Numerical examples show the performance of the method and of an adaptive algorithm for the control of the discretization error

Cut finite element methods for coupled bulk–surface problems

We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and L2L2 norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach

A stable cut finite element method for partial differential equations on surfaces: The Helmholtz–Beltrami operator

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 hk<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 H1 and L2-norms follow

A Cut Finite Element Method for the Bernoulli Free Boundary Value Problem

We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms is added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the $H^1$ Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the the velocity field in the $H^1$ norm. Finally, we present illustrating numerical results

A Cut Finite Element Method for the Bernoulli Free Boundary Value Problem

We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms is added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the $H^1$ Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the the velocity field in the $H^1$ norm. Finally, we present illustrating numerical results

A Stabilized Cut Streamline Diffusion Finite Element Method for Convection-Diffusion Problems on Surfaces

We develop a stabilized cut finite element method for the stationary convection diffusion problem on a surface embedded in R d . The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included

A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator

We consider solving the Laplace–Beltrami problem 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