arXiv subjects

Sara Zahedi

Publications and source records attributed to Sara Zahedi.

At least 19 recordsLinked to original sources

arXiv2026

High-Order Discontinuous Cut Finite Element Methods for Scalar Hyperbolic Conservation Laws

In this paper, we present a family of high-order cut finite element methods based on the discontinuous Galerkin (DG) framework for scalar hyperbolic conservation laws on complex domains. Building on our previous work, we develop a multidimensional formulation that combines macro-element stabilization with flux limiting to obtain a scheme that preserves the maximum principle and remains robust with respect to arbitrary boundary cuts of the background mesh. The physical domain is embedded in a regular background mesh, which may produce arbitrarily small cut cells. To avoid the severe time step restrictions typically associated with such cells, ghost penalty stabilization terms are added on interior facets of macro-elements. The resulting method exhibits stability and accuracy properties similar to those of standard DG methods on fitted meshes. An $L^2$-stability result is derived for the semi-discrete scheme under both periodic and inflow-outflow boundary conditions. To enforce the maximum principle and suppress nonphysical oscillations, we adapt limiter techniques from standard DG methods to the CutFEM setting by defining limiting parameters on macro-elements. In particular, we present a macro-element-based parameterized flux limiter together with adaptations of the Zhang-Shu bound-preserving limiter and the Barth-Jespersen slope limiter. Numerical experiments in two and three spatial dimensions demonstrate optimal convergence orders, preservation of the maximum principle, and accurate shock capturing without spurious oscillations, even for challenging cut configurations involving very small-cut cell intersections.

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arXiv2026

Accelerating droplet-laden Stokes flow simulations with hierarchical surrogate modeling

We present a surrogate modeling strategy for Stokes flows with liquid droplets suspended in a carrier fluid. Our approach is based on a multi-fidelity framework. At the lowest fidelity, droplets are treated as passive tracers, neglecting their influence on the ambient flow field. Building on this approximation, we derive a PDE that represents the current modeling error. This error equation is then solved approximately to correct the flow field and the procedure is iterated. Two fidelities are employed in an alternating fashion: Stokes flow in the absence of droplets and flow around a single droplet in free space. By systematically combining these models, the method captures droplet-flow, droplet-boundary, and droplet-droplet interactions. For geometrically similar droplets, we further develop an efficient offline-online strategy that exploits this structure by reusing precomputed single-droplet solutions. Numerical experiments demonstrate the accuracy and efficiency of the proposed surrogate in a variety of tests, including scenarios with up to 10,000 droplets. Notably, we show that the proposed surrogate achieves substantially reduced computational cost compared to fully resolved multi-fluid simulations with state-of-the-art software.

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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.

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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.

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arXiv2024

A bound preserving cut discontinuous Galerkin method for one dimensional hyperbolic conservation laws

In this paper we present a family of high order cut finite element methods with bound preserving properties for hyperbolic conservation laws in one space dimension. The methods are based on the discontinuous Galerkin framework and use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. Our methods include ghost penalty stabilization to handle small cut elements and a new reconstruction of the approximation on macro-elements, which are local patches consisting of cut and un-cut neighboring elements that are connected by stabilization. We show that the reconstructed solution retains conservation and order of convergence. Our lowest-order scheme results in a piecewise constant solution that satisfies a maximum principle for scalar hyperbolic conservation laws. When the lowest order scheme is applied to the Euler equations, the scheme is positivity preserving in the sense that positivity of pressure and density are retained. For the high-order schemes, suitable bound preserving limiters are applied to the reconstructed solution on macro-elements. In the scalar case, a maximum principle limiter is applied, which ensures that the limited approximation satisfies the maximum principle. Correspondingly, we use a positivity preserving limiter for the Euler equations and show that our scheme is positivity preserving. In the presence of shocks, additional limiting is needed to avoid oscillations, hence we apply a standard TVB limiter to the reconstructed solution. The time step restrictions are of the same order as for the corresponding discontinuous Galerkin methods on the background mesh. Numerical computations illustrate accuracy, bound preservation, and shock capturing capabilities of the proposed schemes.

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arXiv2024

A High-Order Conservative Cut Finite Element Method for Problems in Time-Dependent Domains

A mass-conservative high-order unfitted finite element method for convection-diffusion equations in evolving domains is proposed. The space-time method presented in [P. Hansbo, M. G. Larson, S. Zahedi, Comput. Methods Appl. Mech. Engrg. 307 (2016)] is extended to naturally achieve mass conservation by utilizing Reynold's transport theorem. Furthermore, by partitioning the time-dependent domain into macroelements, a more efficient stabilization procedure for the cut finite element method in time-dependent domains is presented. Numerical experiments illustrate that the method fulfills mass conservation, attains high-order convergence, and the condition number of the resulting system matrix is controlled while sparsity is increased. Problems in bulk domains as well as coupled bulk-surface problems are considered.

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arXiv2023

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.

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arXiv2022

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.

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arXiv2022

High order discontinuous cut finite element methods for linear hyperbolic conservation laws with an interface

We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time-stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.

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arXiv2022

A Cut Finite Element Method for two-phase flows with insoluble surfactants

We propose a new unfitted finite element method for simulation of two-phase flows in presence of insoluble surfactant. The key features of the method are 1) discrete conservation of surfactant mass; 2) the possibility of having meshes that do not conform to the evolving interface separating the immiscible fluids; 3) accurate approximation of quantities with weak or strong discontinuities across evolving geometries such as the velocity field and the pressure. The new discretization of the incompressible Navier--Stokes equations coupled to the convection-diffusion equation modeling the surfactant transport on evolving surfaces is based on a space-time cut finite element formulation with quadrature in time and a stabilization term in the weak formulation that provides function extension. The proposed strategy utilize the same computational mesh for the discretization of the surface Partial Differential Equation (PDE) and the bulk PDEs and can be combined with different techniques for representing and evolving the interface, here the level set method is used. Numerical simulations in both two and three space dimensions are presented including simulations showing the role of surfactant in the interaction between two drops.

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arXiv2021

Conservative Discontinuous Cut Finite Element Methods

We develop a conservative cut finite element method for an elliptic coupled bulk-interface problem. The method is based on a discontinuous Galerkin framework where stabilization is added in such a way that we retain conservation on macro elements containing one element with a large intersection with the domain and possibly a number of elements with small intersections. We derive error estimates and present confirming numerical results

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arXiv2018

A cut finite element method for incompressible two-phase Navier-Stokes flows

We present a space-time Cut Finite Element Method (CutFEM) for the time-dependent Navier-Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. The numerical method is able to accurately capture the strong discontinuity in the pressure and the weak discontinuity in the velocity field across evolving interfaces without re-meshing processes or regularization of the problem. We combine the strategy proposed in [P. Hansbo, M. G. Larson, S. Zahedi, Appl. Numer. Math. 85 (2014), 90--114] for the Stokes equations with a stationary interface and the space-time strategy presented in [P. Hansbo, M. G. Larson, S. Zahedi, Comput. Methods Appl. Mech. Engrg. 307 (2016), 96--116]. We also propose a strategy for computing high order approximations of the surface tension force by computing a stabilized mean curvature vector. The presented space-time CutFEM uses a fixed mesh but includes stabilization terms that control the condition number of the resulting system matrix independently of the position of the interface, ensure stability and a convenient implementation of the space-time method based on quadrature in time. Numerical experiments in two and three space dimensions show that the numerical method is able to accurately capture the discontinuities in the pressure and the velocity field across evolving interfaces without requiring the mesh to be conformed to the interface and with good stability properties.

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arXiv2018

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 ${\mathbb{R}}^d$. 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 $O(h^{-1})$ for a specific choice of method parameters. Numerical example supporting our theoretical results are also included.

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arXiv2017

Stabilization of High Order Cut Finite Element Methods on Surfaces

We develop and analyze a stabilization term for cut finite element approximations of an elliptic second order partial differential equation on a surface embedded in $\mathbb{R}^d$. The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces and provides control of the variation of the finite element solution on the active three dimensional elements that intersect the surface. We show that the condition number of the stiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. The stabilization term works for linear as well as for higher-order elements and the derivation of its stabilizing properties is quite straightforward, which we illustrate by discussing the extension of the analysis to general $n$-dimensional smooth manifolds embedded in $\mathbb{R}^d$, with codimension $d-n$. We also formulate properties of a general stabilization term that are sufficient to prove optimal scaling of the condition number and optimal error estimates in energy- and $L^2$-norm. We finally present numerical studies confirming our theoretical results.

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arXiv2017

A Space-Time Cut Finite Element Method with quadrature in time

We consider convection-diffusion problems in time-dependent domains and present a space-time finite element method based on quadrature in time which is simple to implement and avoids remeshing procedures as the domain is moving. The evolving domain is embedded in a domain with fixed mesh and a cut finite element method with continuous elements in space and discontinuous elements in time is proposed. The method allows the evolving geometry to cut through the fixed background mesh arbitrarily and thus avoids remeshing procedures. However, the arbitrary cuts may lead to ill-conditioned algebraic systems. A stabilization term is added to the weak form which guarantees well-conditioned linear systems independently of the position of the geometry relative to the fixed mesh and in addition makes it possible to use quadrature rules in time to approximate the space-time integrals. We review here the space-time cut finite element method presented in [13] where linear elements are used in both space and time and extend the method to higher order elements for problems on evolving surfaces (or interfaces). We present a new stabilization term which also when higher order elements are used controls the condition number of the linear systems from cut finite element methods on evolving surfaces. The new stabilization combines the consistent ghost penalty stabilization [1] with a term controlling normal derivatives at the interface.

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arXiv2016

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 $\mathbb{R}^3$ gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace-Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and $L^2$ norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.

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arXiv2015

Stabilized CutFEM for the Convection Problem on Surfaces

We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove $h^{3/2}$ order convergence in the natural norm associated with the method and that the full gradient enjoys $h^{3/4}$ order of convergence in $L^2$. We also show that the condition number of the stiffness matrix is bounded by $h^{-2}$. Finally, our results are verified by numerical examples.

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arXiv2015

A cut finite element method for coupled bulk-surface problems on time-dependent domains

In this contribution we present a new computational method for coupled bulk-surface problems on time-dependent domains. The method is based on a space-time formulation using discontinuous piecewise linear elements in time and continuous piecewise linear elements in space on a fixed background mesh. The domain is represented using a piecewise linear level set function on the background mesh and a cut finite element method is used to discretize the bulk and surface problems. In the cut finite element method the bilinear forms associated with the weak formulation of the problem are directly evaluated on the bulk domain and the surface defined by the level set, essentially using the restrictions of the piecewise linear functions to the computational domain. In addition a stabilization term is added to stabilize convection as well as the resulting algebraic system that is solved in each time step. We show in numerical examples that the resulting method is accurate and stable and results in well conditioned algebraic systems independent of the position of the interface relative to the background mesh.

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