We propose nodal auxiliary space preconditioners for facet and edge virtual elements of lowest order by deriving discrete regular decompositions on polytopal grids and generalizing the Hiptmair-Xu preconditioner to the virtual element framework. The preconditioner consists of solving a sequence of elliptic problems on the nodal virtual element space, combined with appropriate smoother steps. Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes. Moreover, we observe numerically that the preconditioner is robust on meshes containing elements with high aspect ratios.

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs RTk \times Qk, k \geq 0. Here Qk is the space of discontinuous polynomial functions of degree less than or equal to k and RT is the Raviart-Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that with the new stabilization term the proposed cut finite element discretization results in pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on RTk \times Qk, k \geq 0. In addition, by decomposing the computational mesh into macroelements and applying ghost penalty terms only on interior edges of macroelements, stabilization is applied very restrictively and active only where needed. Numerical experiments with element pairs RT0 \times Q0, RT1 \times Q1, and BDM1 \times Q0 (where BDM is the Brezzi-Douglas-Marini space) indicate that with the new method we have (1) optimal rates of convergence of the approximate velocity and pressure; (2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; (3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh.

We develop two unfitted finite element methods for the Stokes equations based on \(\textbf{H}^{{{\,\textrm{div}\,}}}\)-conforming finite elements. Both cut finite element methods exhibit optimal convergence order for the velocity, pointwise divergence-free velocity fields, and well-posed linear systems, independently of the position of the boundary relative to the computational mesh. The first method is a cut finite element discretization of the Stokes equations based on the Brezzi–Douglas–Marini (BDM) elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a cut finite element discretization of a three-field formulation of the Stokes problem involving the vorticity, velocity, and pressure and uses the Raviart–Thomas (RT) space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the resulting discrete problems are stable and the divergence-free property of the \(\textbf{H}^{{{\,\textrm{div}\,}}}\)-conforming elements is preserved also on unfitted meshes. In both methods boundary conditions are imposed weakly. We show that imposing Dirichlet boundary conditions weakly introduces additional challenges; (1) The divergence-free property of the RT and the BDM finite elements may be lost depending on how the normal component of the velocity field at the boundary is imposed. (2) Pressure robustness is affected by how well the boundary condition is satisfied and may not hold even if the incompressibility condition holds pointwise. We study two approaches of weakly imposing the normal component of the velocity at the boundary; we either use a penalty parameter and Nitsche’s method or a Lagrange multiplier method. We show that appropriate conditions on the velocity space has to be imposed when Nitsche’s method or penalty is used. Pressure robustness can hold with both approaches by reducing the error at the boundary but the price we pay is seen in the condition numbers of the resulting linear systems, independent of if the mesh is fitted or unfitted to the boundary.

Flat tori are among the only types of Riemannian manifolds forwhich the Laplace eigenvalues can be explicitly computed. In 1964, Milnorused a construction of Witt to find an example of isospectral nonisometricRiemannian manifolds, a striking and concise result that occupied one page inthe Proceedings of the National Academy of Science of the USA. Milnor’s example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues areidentical, in spite of the fact that these tori are not isometric. A natural question is, What is the lowest dimension in which such isospectral nonisometricpairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore thisquestion from all three perspectives and describe its resolution by Schiemannin the 1990s. Moreover, we share a number of open problems.

In this work, we study three different finite element discretizations of the time-harmonic Maxwell’s equations in a bounded domain. These discretizations include the standard edge-based approach, the mixed Kikuchi formulation, and a 3-field formulation based on the finite element exterior calculus framework. We also develop cut finite element versions of each formulation. The unfitted methods are able to retain the properties of the standard fitted methods through the use of mixed stabilization terms. We compare the methods in the fitted and unfitted framework through a source problem and an eigenvalue problem.

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs RT0 × P0, BDM1 × P0, and RT1 × P1. We show that the standard ghost penalty stabilization, often added in the weak forms of Cut Finite Element Methods (CutFEM) for stability and control of the condition number of the resulting linear system matrix, pollutes the computed velocity field so the optimal approximation of the divergence is lost. Therefore, we propose two corrections to the standard stabilization strategy; using macro-elements and new stabilization terms for the pressure. By decomposing the computational mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements stabilization is active only where needed. By modifying the standard stabilization terms for the pressure we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. Numerical experiments indicate that with the new stabilization terms the unfitted finite element discretization results in 1) optimal rates of convergence of the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as for fitted finite element discretizations; 3) optimal approximation of the divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative the computational mesh.

We propose nodal auxiliary space preconditioners for facet and edge virtual elements of lowest order by deriving discrete regular decompositions on polytopal grids and generalizing the Hiptmair-Xu preconditioner to the virtual element framework. The preconditioner consists of solving a sequence of elliptic problems on the nodal virtual element space, combined with appropriate smoother steps. Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes. Moreover, we observe numerically that the preconditioner is robust on meshes containing elements with high aspect ratios.

We extend the divergence preserving cut finite element method presented in [T. Frachon, P. Hansbo, E. Nilsson, S. Zahedi, SIAM J. Sci. Comput., 46 (2024)] for the Darcy interface problem to unfitted outer boundaries. We impose essential boundary conditions on unfitted meshes with a stabilized Lagrange multiplier method. The stabilization term for the Lagrange multiplier is important for stability but it may perturb the approximate solution at the boundary. We study different stabilization terms from cut finite element discretizations of surface partial differential equations and trace finite element methods. To reduce the perturbation we use a Lagrange multiplier space of higher polynomial degree compared to previous work on unfitted discretizations. We propose a symmetric method that results in 1) optimal rates of convergence for the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as for fitted finite element discretizations; 3) optimal approximation of the divergence with pointwise divergence-free approximations of solenoidal velocity fields. The three properties are proven to hold for the lowest order discretization and numerical experiments indicate that these properties continue to hold also when higher order elements are used.