| 61. |
- Logg, Anders, 1976
(författare)
-
Multi-adaptive Galerkin methods for ODEs I
- 2003
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 24:6, s. 1879-1902
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Tidskriftsartikel (refereegranskat)abstract
- We present multi-adaptive versions of the standard continuous and discontinuous Galerkin methods for ODEs. Taking adaptivity one step further, we allow for individual time-steps, order and quadrature, so that in particular each individual component has its own time-step sequence. This paper contains a description of the methods, an analysis of their basic properties, and a posteriori error analysis. In the accompanying paper [A. Logg, SIAM J. Sci. Comput., submitted], we present adaptive algorithms for time-stepping and global error control based on the results of the current paper.
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| 62. |
- Logg, Anders, 1976
(författare)
-
Multi-adaptive Galerkin methods for ODEs II: Implementation and applications
- 2003
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 25:4, s. 1119-1141
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Tidskriftsartikel (refereegranskat)abstract
- Continuing the discussion of the multi-adaptive Galerkin methods mcG(q) and mdG(q) presented in [A. Logg, SIAM J. Sci. Comput., 24 (2003), pp. 1879-1902], we present adaptive algorithms for global error control, iterative solution methods for the discrete equations, features of the implementation Tanganyika, and computational results for a variety of ODEs. Examples include the Lorenz system, the solar system, and a number of time-dependent PDEs.
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| 63. |
- Logg, Anders, 1976
(författare)
-
Multi-adaptive time integration
- 2004
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Ingår i: Applied Numerical Mathematics. - : Elsevier BV. - 0168-9274. ; 48:3-4, s. 339-354
-
Tidskriftsartikel (refereegranskat)abstract
- Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are localised to different components, corresponding to localisation in space for a PDE, efficient time integration thus requires that we use different time steps for different components. We present an overview of the multi-adaptive Galerkin methods mcG(q) and mdG(q) recently introduced in a series of papers by the author. In these methods, the time step sequence is selected individually and adaptively for each component, based on an a posteriori error estimate of the global error. The multi-adaptive methods require the solution of large systems of nonlinear algebraic equations which are solved using explicit-type iterative solvers (fixed point iteration). If the system is stiff, these iterations may fail to converge, corresponding to the well-known fact that standard explicit methods are inefficient for stiff systems. To resolve this problem, we present an adaptive strategy for explicit time integration of stiff ODEs, in which the explicit method is adaptively stabilised by a small number of small, stabilising time steps. © 2003 IMACS. Publised by Elsevier B.V. All rights reserved.
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| 64. |
- Logg, Anders, 1976
(författare)
-
Multiadaptive Galerkin methods for ODEs III: A priori error estimates
- 2006
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 43:6, s. 2624-2646
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Tidskriftsartikel (refereegranskat)abstract
- The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q) for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution. In this paper, we prove general order a priori error estimates for the mcG(q) and mdG(q) methods. To prove the error estimates, we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution. The estimates then follow from interpolation estimates, together with stability estimates for the discrete dual solution. © 2006 Society for Industrial and Applied Mathematics.
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| 65. |
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| 66. |
- Logg, Anders, 1976, et al.
(författare)
-
Solving Poisson’s Equation on the Microsoft HoloLens
- 2017
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Ingår i: Proceedings of the 23rd ACM Symposium on Virtual Reality Software and Technology. - New York, NY, USA : ACM. - 9781450355483 ; Part F131944
-
Konferensbidrag (refereegranskat)abstract
- We present a mixed reality application (HoloFEM) for the Microsoft HoloLens. The application lets a user define and solve a physical problem governed by Poisson's equation with the surrounding real world geometry as input data. Holograms are used to visualise both the problem and the solution. The finite element method is used to solve Poisson's equation. Solving and visualising partial differential equations in mixed reality could have potential usage in areas such as building planning and safety engineering.
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| 67. |
- Massing, André, et al.
(författare)
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A Nitsche-Based Cut Finite Element Method for a Fluid--Structure Interaction Problem
- 2015
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Ingår i: Communications in Applied Mathematics and Computational Science. - : Mathematical Sciences Publishers. - 1559-3940 .- 2157-5452. ; 10:2, s. 97-120
-
Tidskriftsartikel (refereegranskat)abstract
- We present a new composite mesh finite element method for fluid-structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh that is embedded into a fixed background fluid mesh. The embedding allows for an arbitrary overlap of the fluid meshes. The coupling between the embedded and background fluid meshes is enforced using a stabilized Nitsche formulation that allows us to establish stability and optimal-order a priori error estimates. We consider here a steady state fluid-structure interaction problem where a hyperelastic structure interacts with a viscous fluid modeled by the Stokes equations. We evaluate an iterative solution procedure based on splitting and present three-dimensional numerical examples.
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| 68. |
- Massing, Andre, et al.
(författare)
-
A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem
- 2014
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 61:3, s. 604-628
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Tidskriftsartikel (refereegranskat)abstract
- We present a novel finite element method for the Stokes problem on fictitious domains. We prove inf-sup stability, optimal order convergence and uniform boundedness of the condition number of the discrete system. The finite element formulation is based on a stabilized Nitsche method with ghost penalties for the velocity and pressure to obtain stability in the presence of small cut elements. We demonstrate for the first time the applicability of the Nitsche fictitious domain method to three-dimensional Stokes problems. We further discuss a general, flexible and freely available implementation of the method and present numerical examples supporting the theoretical results.
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| 69. |
- Massing, Andre, et al.
(författare)
-
A stabilized Nitsche overlapping mesh method for the Stokes problem
- 2014
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 128:1, s. 73-101
-
Tidskriftsartikel (refereegranskat)abstract
- We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By extending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results.
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| 70. |
- Massing, Andre, et al.
(författare)
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Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions
- 2013
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Ingår i: SIAM Journal of Scientific Computing. - 1064-8275 .- 1095-7197 .- 1095-7200. ; 35:1, s. C23-C47
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Tidskriftsartikel (refereegranskat)abstract
- In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on nonmatching or overlapping meshes. Examples of such methods are the fictitious domain method, the extended finite element method, and Nitsche's method. In all these methods, integrals must be computed over cut cells or subsimplices, which is challenging to implement, especially in three space dimensions. In this note, we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices. As a canonical example of an overlapping mesh method, we consider Nitsche's method, which we apply to Poisson's equation and a linear elastic problem. © 2013 Society for Industrial and Applied Mathematics.
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