| 1. |
- Burman, Erik, et al.
(author)
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CutFEM based on extended finite element spaces
- 2022
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In: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 152, s. 331-369
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Journal article (peer-reviewed)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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| 2. |
- Burman, Erik, et al.
(author)
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Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions
- 2024
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In: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 93:345, s. 35-54
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Journal article (peer-reviewed)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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| 3. |
- Hansbo, Peter, et al.
(author)
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STABILIZED FINITE ELEMENT APPROXIMATION OF THE MEAN CURVATURE VECTOR ON CLOSED SURFACES
- 2015
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In: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 53:4, s. 1806-1832
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Journal article (peer-reviewed)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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| 4. |
- Burman, Erik, et al.
(author)
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A cut discontinuous Galerkin method for the Laplace–Beltrami operator
- 2017
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In: IMA Journal of Numerical Analysis. - : Oxford University Press. - 0272-4979 .- 1464-3642. ; 37:1, s. 138-169
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Journal article (peer-reviewed)abstract
- We develop a discontinuous cut finite element method for the Laplace–Beltrami operator on a hypersurface embedded in R. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in R. The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then, a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighbouring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.
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| 5. |
- Burman, Erik, et al.
(author)
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A cut finite element method for the Bernoulli free boundary value problem
- 2017
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In: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 317, s. 598-618
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Journal article (peer-reviewed)abstract
- We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H1 Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H1norm. Finally, we present illustrating numerical results.
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| 6. |
- Burman, Erik, et al.
(author)
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Augmented Lagrangian finite element methods for contact problems
- 2019
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In: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 53:1, s. 173-195
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Journal article (peer-reviewed)abstract
- We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution.
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| 7. |
- Burman, Erik, et al.
(author)
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The Penalty-Free Nitsche Method and Nonconforming Finite Elements for the Signorini Problem
- 2017
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In: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 55:6, s. 2523-2539
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Journal article (peer-reviewed)abstract
- We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild [SIAM J. Numer. Anal., 51 ( 2013), pp. 1295-1307], our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm.
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| 8. |
- Hansbo, Peter, 1959-, et al.
(author)
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A stabilized finite element method for the Darcy problem on surfaces
- 2017
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In: IMA Journal of Numerical Analysis. - : Oxford University Press. - 0272-4979 .- 1464-3642. ; 37:3, s. 1274-1299
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Journal article (peer-reviewed)abstract
- We consider a stabilized finite element method for the Darcy problem on a surface based on the Masud–Hughes formulation. A special feature of the method is that the tangential condition of the velocity field is weakly enforced through the bilinear form, and that standard parametric continuous polynomial spaces on triangulations can be used. We prove optimal order a priori estimates that take the approximation of the geometry and the solution into account.
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| 9. |
- Abrahamsson, Fredrik
(author)
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Strong L1 convergence to equilibrium without entropy conditions for the Boltzmann equation
- 1999
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In: Communications in Partial Differential Equations. - 0360-5302 .- 1532-4133. ; 24:7-8, s. 1501-1535
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Journal article (peer-reviewed)abstract
- The main result of this paper is that for the har dsphere kernel, the solution of the spatially homogenous Boltzmann equation converges strongly in L1 to equilibrium given that the initial data f0 belongs to L1(R3,(1+v^2)dv). This was previously known to be true with the additional assumption that f0logf0 belonged to L1(R3), which corresponds to bounded initial entropy.
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| 10. |
- Andersson, Anders, 1957-
(author)
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On the curvature of an inner curve in a Schwarz--Christoffel mapping
- 2009
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In: Further Progress in Analysis. - : World Scientific. - 9789812837325 - 9812837329 ; , s. 281-290
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Conference paper (peer-reviewed)abstract
- In the so called outer polygon method, an approximative conformal mapping for a given simply connected region Ω is constructed using a Schwarz–Christoffel mapping for an outer polygon, a polygonal region of which Ω is a subset. The resulting region is then bounded by a C∞-curve, which among other things means that its curvature is bounded.In this work, we study the curvature of an inner curve in a polygon, i.e., the image under the Schwarz–Christoffel mapping from R, the unit disk or upper half–plane, to a polygonal region P of a curve inside R. From the Schwarz–Christoffel formula, explicit expressions for the curvature are derived, and for boundary curves, appearing in the outer polygon method, estimations of boundaries for the curvature are given.
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