arXiv subjects

Erik Nilsson

Publications and source records attributed to Erik Nilsson.

8 recordsLinked to original sources

arXiv2026

Ghost stabilisation for cut finite element exterior calculus

We introduce the cut finite element method in the language of finite element exterior calculus, by formulating a stabilisation -- for any form degree -- that makes the method robust with respect to the position of the interface relative to the mesh. We prove that the $L^2$-norm on the physical domain augmented with this stabilisation is uniformly equivalent to the $L^2$-norm on the ``active'' mesh that contains all the degrees of freedom of the finite element space (including those external to the physical domain). We show how this CutFEEC method can be applied to discretize the Hodge Laplace equations on an unfitted mesh, in any dimension and any topology, and prove stability and optimal convergence. Two numerical examples are provided, with convergence and condition number scaling independent of the position of the boundary with respect to the background mesh. We first solve the Hodge Laplace equation on a conforming finite element space of $H^{\text{curl}}$ posed on a filled torus. The second example extends beyond the Hodge Laplace problem and illustrates the importance of stabilisation for pressure robustness in unfitted schemes, using an $H^{\text{curl}}$-formulation for the Stokes equations.

math.NA↗

arXiv2026

Divergence-free unfitted finite element discretisations for the Darcy problem

We develop an unfitted compatible finite element discretisation for the Darcy problem based on $H(\mathrm{div})$-conforming flux spaces and discontinuous pressure spaces. The method is designed to preserve pointwise discrete mass conservation while remaining robust in the presence of arbitrarily small cut cells arising from unfitted meshes. Robustness is achieved by combining an $L^2$-stabilisation of the flux with an additional mixed-term stabilisation that enhances pressure control without destroying the local conservation structure. We consider both cell-wise (bulk) and face-based ghost-penalty realisations of the stabilisation. Mixed boundary conditions are handled by weak imposition of both flux and pressure traces on unfitted boundaries. We prove stability and a priori error estimates with constants independent of the cut configuration, and establish pressure-robust flux error bounds in the case of pure pressure boundary conditions. We also introduce an augmented Lagrangian variant that improves control of the conservation constraint and is amenable to efficient preconditioning strategies. Numerical experiments for a range of cut configurations, boundary-condition regimes and parameter choices confirm the theoretical results, demonstrating optimal convergence, cut-independent conditioning and mass conservation up to solver tolerance.

math.NA↗

arXiv2026

Uniformly Bounded Cochain Extensions and Uniform Poincaré Inequalities

In this paper, we construct a novel global bounded cochain extension operator for differential forms on Lipschitz domains. Building upon the classical universal extension of Hiptmair, Li, and Zou, our construction restores global commutativity with the exterior derivative in the natural $HΛ^k(Ω)$ setting. The construction applies to domains and ambient extension sets of arbitrary topology, with strict commutation holding on the orthogonal complement of harmonic forms, as dictated by the underlying topological obstruction. This provides a missing analytical tool for the rigorous foundation of Cut Finite Element Methods (CutFEM). We also obtain continuous uniform Poincaré inequalities and lower bounds for the first Neumann eigenvalue on non-convex domains.

math.FA↗

arXiv2024

Nodal auxiliary space preconditioners for mixed virtual element methods

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.

math.NA↗

arXiv2024

Divergence-free cut finite element methods for Stokes flow

We develop two unfitted finite element methods for the Stokes equations using $H^{\text{div}}$-conforming finite elements. Both methods achieve optimal convergence for velocity, ensure pointwise divergence-free velocity fields, and produce well-posed linear systems, regardless of the boundary's position relative to the computational mesh. The first method is a cut finite element discretization of the Stokes equations based on Brezzi-Douglas-Marini (BDM) elements, incorporating interior penalty terms to enforce tangential continuity of velocity at interior mesh edges. The second method involves a cut finite element discretization of a three-field formulation of the Stokes problem, utilizing Raviart-Thomas (RT) space for velocity. We introduce mixed ghost penalty stabilization terms for both methods to ensure stability and to preserve the divergence-free property of the $H^{\text{div}}$-conforming elements, even on unfitted meshes. Boundary conditions in both methods are imposed weakly, which presents challenges: 1) The divergence-free property of the RT and BDM finite elements may be compromised depending on how the normal component of the velocity field at the boundary is imposed. 2) Pressure robustness is influenced by the accuracy of boundary condition enforcement and may fail even if the incompressibility condition holds pointwise. We explore two approaches for weakly imposing the normal component of the boundary velocity: using a penalty parameter with Nitsche's method or a Lagrange multiplier method. We demonstrate that specific conditions on the velocity space are necessary when employing Nitsche's method or penalty. While pressure robustness can be maintained with both approaches by minimizing boundary errors, this comes at the cost of increased condition numbers in the resulting linear systems, whether the mesh is fitted or unfitted to the boundary.

math.NA↗

arXiv2024

Stabilized Lagrange Multipliers for Dirichlet Boundary Conditions in Divergence Preserving Unfitted Methods

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.

math.NA↗

arXiv2024

A divergence preserving cut finite element method for Darcy flow

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_k\times Q_k$, $k\geq 0$. Here $Q_k$ is the space of discontinuous polynomial functions of degree less or equal to $k$ and $\textbf{RT}$ 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 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 the method with the new stabilization term has pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on $\textbf{RT}_k\times Q_k$, $k\geq 0$. In addition, by decomposing the mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements, stabilization is applied very restrictively and only where needed. Numerical experiments with element pairs $\textbf{RT}_0\times Q_0$, $\textbf{RT}_1\times Q_1$, and $\textbf{BDM}_1\times Q_0$ (where $\textbf{BDM}$ is the Brezzi-Douglas-Marini space) indicate that 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.

math.NA↗

arXiv2022

The isospectral problem for flat tori from three perspectives

Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral non-isometric Riemannian manifolds, a striking and concise result that occupied one page in the 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 are identical, in spite of the fact that these tori are not isometric. A natural question is: what is the \em lowest \em dimension in which such isospectral non-isometric pairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s. Moreover, we share a number of open problems.

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