| 1. |
- Hansbo, Peter F G, 1959, et al.
(författare)
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An adaptive low-order FE-scheme for Stokes flow with cavitation
- 2010
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Ingår i: Journal of Numerical Mathematics. - 1570-2820 .- 1569-3953. ; 18:3, s. 177-185
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Tidskriftsartikel (refereegranskat)abstract
- In this note we derive a posteriori error bounds for FE-discretisations for a fluid problem with cavitation. The underlying model is the Stokes system together with an inequality constraint for the pressure. In order to avoid suboptimal behavior of the error bounds we propose to employ a Lagrange setting yielding an improved estimate. Numerical tests confirm our theoretical results.
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| 2. |
- Hansbo, Anita, 1960-, et al.
(författare)
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A finite element method on composite grids based on Nitsche's method
- 2003
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 37:3, s. 495-514
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose a finite element method for the approximation of second order elliptic problems on composite grids. The method is based on continuous piecewise polynomial approximation on each grid and weak enforcement of the proper continuity at an artificial interface defined by edges (or faces) of one the grids. We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions, and present some numerical examples.
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| 3. |
- Hansbo, Anita, 1960-, et al.
(författare)
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An unfitted finite element method, based on Nitsche's method, for elliptic interface problems
- 2002
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - 0045-7825 .- 1879-2138. ; 191:47-48, s. 5537-5552
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose a method for the finite element solution of elliptic interface problem, using an approach due to Nitsche. The method allows for discontinuities, internal to the elements, in the approximation across the interface. We show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh. Further, we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples.
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| 4. |
- Hansbo, Peter, et al.
(författare)
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A cut finite element method for a Stokes interface problem
- 2014
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Ingår i: Applied Numerical Mathematics. - : Elsevier BV. - 0168-9274 .- 1873-5460. ; 85, s. 90-114
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Tidskriftsartikel (refereegranskat)abstract
- We present a finite element method for the Stokes equations involving two immiscible incompressible fluids with different viscosities and with surface tension. The interface separating the two fluids does not need to align with the mesh. We propose a Nitsche formulation which allows for discontinuities along the interface with optimal a priori error estimates. A stabilization procedure is included which ensures that the method produces a well conditioned stiffness matrix independent of the location of the interface.
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| 5. |
- Burman, Erik, et al.
(författare)
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CutFEM based on extended finite element spaces
- 2022
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 152, s. 331-369
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Tidskriftsartikel (refereegranskat)abstract
- We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More, precisely the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.
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| 6. |
- Burman, Erik, et al.
(författare)
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Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions
- 2024
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 93:345, s. 35-54
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Tidskriftsartikel (refereegranskat)abstract
- We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution u ∈ Hs with s ∈ (1, 3/2]. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.
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| 7. |
- Hansbo, Peter, et al.
(författare)
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STABILIZED FINITE ELEMENT APPROXIMATION OF THE MEAN CURVATURE VECTOR ON CLOSED SURFACES
- 2015
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 53:4, s. 1806-1832
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Tidskriftsartikel (refereegranskat)abstract
- The mean curvature vector of a surface is obtained by letting the Laplace-Beltrami operator act on the embedding of the surface in R-3. In this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in L-2. The stabilization involves the jump in the tangent gradient in the direction of the outer co-normals at each edge in the surface mesh. We consider both standard meshed surfaces and so-called cut surfaces that are level sets of piecewise linear distance functions. We prove a priori error estimates and verify the theoretical results numerically.
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| 8. |
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| 9. |
- Hansbo, Peter, et al.
(författare)
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A simple nonconforming tetrahedral element for the Stokes equations
- 2022
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 400
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we apply a nonconforming rotated bilinear tetrahedral element to the Stokes problem in R3. We show that the element is stable in combination with a piecewise linear, continuous, approximation of the pressure. This gives an approximation similar to the well known continuous P2–P1 Taylor–Hood element, but with fewer degrees of freedom. The element is a stable non-conforming low order element which fulfils Korn's inequality, leading to stability also in the case where the Stokes equations are written on stress form for use in the case of free surface flow.
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| 10. |
- Yedeg, Esubalewe Lakie, 1983-, et al.
(författare)
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A Nitsche-type Method for Helmholtz Equation with an Embedded Acoustically Permeable Interface
- 2016
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 304, s. 479-500
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Tidskriftsartikel (refereegranskat)abstract
- We propose a new finite element method for Helmholtz equation in the situation where an acoustically permeable interface is embedded in the computational domain. A variant of Nitsche's method, different from the standard one, weakly enforces the impedance conditions for transmission through the interface. As opposed to a standard finite-element discretization of the problem, our method seamlessly handles a complex-valued impedance function Z that is allowed to vanish. In the case of a vanishing impedance, the proposed method reduces to the classic Nitsche method to weakly enforce continuity over the interface. We show stability of the method, in terms of a discrete Gårding inequality, for a quite general class of surface impedance functions, provided that possible surface waves are sufficiently resolved by the mesh. Moreover, we prove an a priori error estimate under the assumption that the absolute value of the impedance is bounded away from zero almost everywhere. Numerical experiments illustrate the performance of the method for a number of test cases in 2D and 3D with different interface conditions.
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