| 1. |
- Wadbro, Eddie, 1981-, et al.
(författare)
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A uniformly well-conditioned, unfitted Nitsche method for interface problems
- 2013
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Ingår i: BIT Numerical Mathematics. - : Springer. - 0006-3835 .- 1572-9125. ; 53:3, s. 791-820
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Tidskriftsartikel (refereegranskat)abstract
- A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented. The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and jump conditions at the interface are weakly enforced using a variant of Nitsche’s method. In our method, the solutions on each side of the interface are extended to the entire domain which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. We prove that the method provides optimal convergence order in the energy and the L2 norms and a condition number of the system matrix that is independent of the position of the interface relative to the mesh. Numerical experiments confirm the theoretical results and demonstrate optimal convergence order also for the pointwise errors.
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| 2. |
- Burman, Erik, et al.
(författare)
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A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 358
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Tidskriftsartikel (refereegranskat)abstract
- We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. 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.
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| 3. |
- Burman, Erik, et al.
(författare)
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Stabilized CutFEM for the convection problem on surfaces
- 2019
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Ingår i: Numerische Mathematik. - : Springer Berlin/Heidelberg. - 0029-599X .- 0945-3245. ; 141:1, s. 103-139
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Tidskriftsartikel (refereegranskat)abstract
- 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 h3/2 order convergence in the natural norm associated with the method and that the full gradient enjoys h3/4 order of convergence in L2. 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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| 4. |
- Frachon, Thomas, et al.
(författare)
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A cut finite element method for incompressible two-phase Navier–Stokes flows
- 2019
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Ingår i: Journal of Computational Physics. - : Academic Press. - 0021-9991 .- 1090-2716. ; 384, s. 77-98
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Tidskriftsartikel (refereegranskat)abstract
- 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 et al. (2014) [14] for the Stokes equations with a stationary interface and the space–time strategy presented in P. Hansbo et al. (2016) [20]. 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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| 5. |
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| 6. |
- Frachon, Thomas, et al.
(författare)
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A cut finite element method for two-phase flows with insoluble surfactants
- 2023
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 473, s. 111734-
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Tidskriftsartikel (refereegranskat)abstract
- 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 utilizes 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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| 7. |
- Frachon, Thomas, et al.
(författare)
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Cut finite element methods for Darcy flow in fractured porous media
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 8. |
- Fu, Pei, et al.
(författare)
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High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface
- 2022
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Ingår i: Journal of Scientific Computing. - : Springer Nature. - 0885-7474 .- 1573-7691. ; 90
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Tidskriftsartikel (refereegranskat)abstract
- 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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| 9. |
- Larson, Mats G., et al.
(författare)
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Conservative cut finite element methods using macroelements
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 414
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Tidskriftsartikel (refereegranskat)abstract
- 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 macroelements 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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| 10. |
- Larson, Mats G., et al.
(författare)
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Stabilization of high order cut finite element methods on surfaces
- 2020
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Ingår i: IMA Journal of Numerical Analysis. - : Oxford University Press. - 0272-4979 .- 1464-3642. ; 40:3, s. 1702-1745
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Tidskriftsartikel (refereegranskat)abstract
- We develop and analyse a stabilization term for cut finite element approximations of an elliptic second-order partial differential equation on a surface embedded in 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 R-d, with codimension d - n. We also state the 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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