| 1. |
- 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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| 2. |
- Burman, E., et al.
(författare)
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Extension operators for trimmed spline spaces
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 403
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Tidskriftsartikel (refereegranskat)abstract
- We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree p with k continuous derivatives. The construction is based on polynomial extension from neighboring elements together with projection back into the spline space. We prove stability and approximation results for the extension operator. Finally, we illustrate how we can use the extension operator to construct a stable cut isogeometric method for an elliptic model problem.
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| 3. |
- Creignou, Nadia, et al.
(författare)
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Paradigms for Parameterized Enumeration
- 2017
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Ingår i: Theory of Computing Systems. - : Springer. - 1432-4350 .- 1433-0490. ; 60:4, s. 737-758
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Tidskriftsartikel (refereegranskat)abstract
- The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First, we define formally different notions of efficient enumeration in the context of parameterized complexity: FPT-enumeration and delayFPT. Second, we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems. Furthermore, we study the parameterized complexity of enumerating all models of Boolean formulas having weight at least k, where k is the parameter, in the famous Schaefer’s framework. We consider propositional formulas that are conjunctions of constraints taken from a fixed finite set Γ. Given such a formula and an integer k, we are interested in enumerating all the models of the formula that have weight at least k. We obtain a dichotomy classification and prove that, according to the properties of the constraint language Γ, either one can enumerate all such models in delayFPT, or no such delayFPT enumeration algorithm exists under some complexity-theoretic assumptions.
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| 4. |
- Burman, Erik, et al.
(författare)
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Explicit time stepping for the wave equation using cutFEM with discrete extension
- 2022
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 44:3, s. A1254-A1289
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples.
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| 5. |
- Burman, Erik N., et al.
(författare)
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A cut finite element method with boundary value correction for the incompressible Stokes equations
- 2019
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Ingår i: Numerical mathematics and advanced applications ENUMATH 2017. - Cham : Springer. - 9783319964140 - 9783319964157 ; , s. 183-192
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Konferensbidrag (refereegranskat)abstract
- We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche’s method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman et al. (Math Comput 87(310):633–657, 2018).
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| 6. |
- Jonsson, Peter, et al.
(författare)
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The exponential-time hypothesis and the relative complexity of optimization and logical reasoning problems
- 2021
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Ingår i: Theoretical Computer Science. - : Elsevier. - 0304-3975 .- 1879-2294. ; 892, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Algebraic techniques introduced by Jonsson et al. (2017) [4] show that the fine-grained time complexity of the parameterized [Formula presented] problem correlates to the lattice of strong partial clones. With this ordering they isolated a relation R such that [Formula presented] can be solved at least as fast as any other NP-hard [Formula presented] problem. In this paper we extend this method and show that such languages also exist for the surjective SAT problem, the max ones problem, the propositional abduction problem, and the Boolean valued constraint satisfaction problem over finite-valued constraint languages. These languages may be interesting when investigating the borderline between polynomial time, subexponential time and exponential-time algorithms since they in a precise sense can be regarded as NP-hard problems with minimum time complexity. Indeed, with the help of these languages we relate all of the above problems to the exponential time hypothesis (ETH) in several different ways.
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| 7. |
- Cenanovic, Mirza, et al.
(författare)
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Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 372
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces. In particular, we adopt a stabilization technique which allows for first order L2-convergence of the mean curvature vector and apply this stabilization technique also to the computation of continuous, recovered, normals using L2-projections of the piecewise constant face normals. Finally, we use our projected normals to define an adaptive mesh refinement approach to geometry resolution where we also employ spline techniques to reconstruct the surface before refinement. We compare our results to previously proposed approaches.
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| 8. |
- Burman, E., et al.
(författare)
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A stable cut finite element method for partial differential equations on surfaces : The Helmholtz–Beltrami operator
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 362
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Tidskriftsartikel (refereegranskat)abstract
- We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions. Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition hk<C, where h denotes the mesh size, k the wave number and C a constant depending mainly on the surface curvature κ, but not on the surface/mesh intersection. Optimal error estimates in the H1 and L2-norms follow.
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