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- Standar, Christoffer, 1988
(författare)
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On finite element schemes for Vlasov-Maxwell system and Schrödinger equation
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis treats finite element schemes for two kind of problems, the Valsov-Maxwellsystem and the nonlinear Schrödinger equation. We study streamline diffusion schemes applied for numerical solution of the one and one-half dimensional relativistic Vlasov-Maxwell system. The study is made both in a priori and a posteriori settings. In the a priori setting we derive stability estimates and prove optimal convergence rates, due to the maximal available regularity of the exact solution. In addition to this we also prove existence and uniqueness of the numerical solution. In the a posteriori setting we use dual problems to prove error estimates in L_\infty (H^{-1}) norm. For the Maxwell equation we also prove error estimates in H^{-1} (H^{-1}) norms. Further more we study a hp-version of the streamline diffusion scheme for thethree dimensional Vlasov-Maxwell system in an a priori setting. A Nitsche type scheme is also introduced and analyzed for Maxwell's equations. For the nonlinear Schrödinger equation a two level time discretization is used. Here we derive a priori error estimates both in L_2 and H^1 norms.
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| 3. |
- Standar, Christoffer, 1988
(författare)
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On streamline diffusion schemes for the one and one-half dimensional relativistic Vlasov-Maxwell system
- 2016
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Ingår i: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 53:2, s. 147-169
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Tidskriftsartikel (refereegranskat)abstract
- We study streamline diffusion schemes applied for numerical solution of the one and one-half dimensional relativistic Vlasov-Maxwell system arising in modeling plasma of particles governed by electro-magnetic fields. We derive stability estimates and prove optimal convergence rates, due to the maximal available regularity of the exact solution.
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| 5. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A posteriori error estimates for streamline-diffusion and discontinuous Galerkin methods for the Vlasov-Maxwell system
- 2018
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 58:1, s. 5-26
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov– Maxwell system. The SD scheme yields a weak formulation, that corresponds to an add of extra diffusion to, e.g. the system of equations having hyperbolic nature, or convection-dominated convection diffusion problems. The a posteriori error estimates rely on dual formulations and yield error controls based on the computable residuals. The convergence estimates are derived in negative norms, where the error is split into an iteration and an approximation error and the iteration procedure is assumed to converge.
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| 6. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A posteriori error estimates for the one and one-half Dimensional Relativistic Vlasov–Maxwell system
- 2018
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 58:1, s. 5-26
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Tidskriftsartikel (refereegranskat)abstract
- © 2017 The Author(s) This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov–Maxwell system. The SD scheme yields a weak formulation, that corresponds to an add of extra diffusion to, e.g. the system of equations having hyperbolic nature, or convection-dominated convection diffusion problems. The a posteriori error estimates rely on dual formulations and yield error controls based on the computable residuals. The convergence estimates are derived in negative norms, where the error is split into an iteration and an approximation error and the iteration procedure is assumed to converge.
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| 7. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A Priori Error Estimates and Computational Studies for a Fermi Pencil-Beam Equation
- 2018
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Ingår i: Journal of Computational and Theoretical Transport. - : Informa UK Limited. - 2332-4325 .- 2332-4309. ; 47:1-3, s. 125-151
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Tidskriftsartikel (refereegranskat)abstract
- We derive a priori error estimates for the standard Galerkin and streamline diffusion finite element methods for the Fermi pencil-beam equation obtained from a fully three-dimensional Fokker-Planck equation in space and velocity variables. For a constant transport cross-section, there is a closed form analytic solution available for the Fermi equation with a data as product of Dirac functions. Our objective is to study the case of nonconstant, nonincreasing transport cross-section. Therefore we start with a theoretical, that is, a priori, error analysis for a Fermi model with modified initial data in L-2. Then we construct semi-streamline-diffusion and characteristic streamline-diffusion schemes and consider an adaptive algorithm for local mesh refinements. To derive the stability estimates, for simplicity, we rely on the assumption of nonincreasing transport cross-section. Different numerical examples, in two space dimensions are justifying the theoretical results. Implementations show significant reduction of the computational error by using such adaptive procedure.
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| 10. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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Convergence of hp-Streamline Diffusion Method for Vlasov-Maxwell System
- 2019
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Ingår i: Journal of Computational and Theoretical Transport. - : Informa UK Limited. - 2332-4309 .- 2332-4325. ; 48:7, s. 263-279
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study stability and convergence for hp-streamline diffusion (SD) finite element method for the, relativistic, time-dependent Vlasov-Maxwell (VM) system. We consider spatial domain and velocities The objective is to show globally optimal a priori error bound of order for the SD approximation of the VM system; where is the mesh size and is the spectral order. Our estimates rely on the local version with h(K) being the diameter of the phase-space-time element K and p(K) the spectral order for K. The optimal hp estimates require an exact solution in the Sobolev space Numerical implementations, performed for examples in one space- and two velocity dimensions, are justifying the robustness of the theoretical results.
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