| 1. |
- Arndt, Daniel, et al.
(författare)
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The deal.II library, Version 9.4
- 2022
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Ingår i: Journal of Numerical Mathematics. - : Walter de Gruyter. - 1570-2820 .- 1569-3953. ; 30:3, s. 231-246
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Tidskriftsartikel (refereegranskat)abstract
- This paper provides an overview of the new features of the finite element library deal.II, version 9.4.
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| 2. |
- Gürkan, Ceren, et al.
(författare)
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Stabilized Cut Discontinuous Galerkin Methods for Advection-Reaction Problems
- 2020
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 42:5, s. A2620-A2654
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Tidskriftsartikel (refereegranskat)abstract
- We develop novel stabilized cut discontinuous Galerkin methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in R-d where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the stationary advection-reaction problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two- and three-dimensional test problems.
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| 3. |
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| 4. |
- Ludvigsson, Gustav, et al.
(författare)
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High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains
- 2018
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 76:2, s. 812-847
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.
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| 5. |
- Schoeder, Svenja, et al.
(författare)
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High-order cut discontinuous Galerkin methods with local time stepping for acoustics
- 2020
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Ingår i: International Journal for Numerical Methods in Engineering. - : John Wiley & Sons. - 0029-5981 .- 1097-0207. ; 121:13, s. 2979-3003
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Tidskriftsartikel (refereegranskat)abstract
- We propose a method to solve the acoustic wave equation on an immersed domain using the hybridizable discontinuous Galerkin method for spatial discretization and the arbitrary derivative method with local time stepping (LTS) for time integration. The method is based on a cut finite element approach of high order and uses level set functions to describe curved immersed interfaces. We study under which conditions and to what extent small time step sizes balance cut instabilities, which are present especially for high-order spatial discretizations. This is done by analyzing eigenvalues and critical time steps for representative cuts. If small time steps cannot prevent cut instabilities, stabilization by means of cell agglomeration is applied and its effects are analyzed in combination with local time step sizes. Based on two examples with general cuts, performance gains of the LTS over the global time stepping are evaluated. We find that LTS combined with cell agglomeration is most robust and efficient.
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| 6. |
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| 7. |
- Sticko, Simon, 1988-
(författare)
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High Order Cut Finite Element Methods for Wave Equations
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis considers wave propagation problems solved using finite element methods where a boundary or interface of the domain is not aligned with the computational mesh. Such methods are usually referred to as cut or immersed methods. The motivation for using immersed methods for wave propagation comes largely from scattering problems when the geometry of the domain is not known a priori. For wave propagation problems, the amount of computational work per dispersion error is generally lower when using a high order method. For this reason, this thesis aims at studying high order immersed methods.Nitsche's method is a common way to assign boundary or interface conditions in immersed finite element methods. Here, penalty terms that are consistent with the boundary/interface conditions are added to the weak form. This requires that special quadrature rules are constructed on the intersected elements, which take the location of the immersed boundary/interface into account. A common problem for all immersed methods is small cuts occurring between the elements in the mesh and the computational domain. A suggested way to remedy this is to add terms penalizing jumps in normal derivatives over the faces of the intersected elements.Paper I and Paper II consider the acoustic wave equation, using first order elements in Paper I, and using higher order elements in Paper II. High order elements are then used for the elastic wave equation in Paper III. Papers I to III all use continuous Galerkin, Nitsche's method, and jump-stabilization. Paper IV compares the errors of this type of cut finite element method with two other numerical methods. One result from Paper II is that the added jump-stabilization results in a mass matrix with a high condition number. This motivates the investigation of alternatives. Paper V considers a hybridizable discontinuous Galerkin method. This paper investigates to what extent local time stepping in combination with cell-merging can be used to overcome the problem of small cuts.
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| 8. |
- Sticko, Simon, et al.
(författare)
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High-order cut finite elements for the elastic wave equation
- 2020
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Ingår i: Advances in Computational Mathematics. - : Springer. - 1019-7168 .- 1572-9044. ; 46:3
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Tidskriftsartikel (refereegranskat)abstract
- A high-order cut finite element method is formulated for solving the elastic wave equation. Both a single domain problem and an interface problem are treated. The boundary or interface is allowed to cut through the background mesh. To avoid problems with small cuts, stabilizing terms are added to the bilinear forms corresponding to the mass and stiffness matrix. The stabilizing terms penalize jumps in normal derivatives over the faces of the elements cut by the boundary/interface. This ensures a stable discretization independently of how the boundary/interface cuts the mesh. Nitsche’s method is used to enforce boundary and interface conditions, resulting in symmetric bilinear forms. As a result of the symmetry, an energy estimate can be made and optimal order a priori error estimates are derived for the single domain problem. Finally, numerical experiments in two dimensions are presented that verify the order of accuracy and stability with respect to small cuts.
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| 9. |
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| 10. |
- Sticko, Simon
(författare)
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Towards higher order immersed finite elements for the wave equation
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We consider solving the scalar wave equation using immersed finite elements. Such a method might be useful, for instance, in scattering problems when the geometry of the domain is not known a priori. For hyperbolic problems, the amount of computational work per dispersion error is generally lower when using higher order methods. This serves as motivation for considering a higher order immersed method.One problem in immersed methods is how to enforce boundary conditions. In the present work, boundary conditions are enforced weakly using Nitsche's method. This leads to a symmetric weak formulation, which is essential when solving the wave equation. Since the discrete system consists of symmetric matrices, having real eigenvalues, this ensures stability of the semi-discrete problem.In immersed methods, small intersections between the immersed domain and the elements of the background mesh make the system ill-conditioned. This ill-conditioning becomes increasingly worse when using higher order elements. Here, we consider resolving this issue using additional stabilization terms. These terms consist of jumps in higher order derivatives acting on the internal faces of the elements intersected by the boundary.
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| 11. |
- Ulfsby, Tale Bakken, et al.
(författare)
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Stabilized cut discontinuous Galerkin methods for advection–reaction problems on surfaces
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 413
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Tidskriftsartikel (refereegranskat)abstract
- We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection–reaction problems on surfaces embedded in Rd. The CutDG method is based on embedding the surface into a full-dimensional background mesh and using the associated discontinuous piecewise polynomials of order k as test and trial functions. As the surface can cut through the mesh in an arbitrary fashion, we design a suitable stabilization that enables us to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. The resulting CutDG formulation is geometrically robust in the sense that all derived theoretical results hold with constants independent of any particular cut configuration. Numerical examples support our theoretical findings.
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