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| 2. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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Convergence of Finite Volume Scheme for a Three-Dimensional Poisson Equation
- 2014
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Ingår i: Journal of Mathematical Sciences. - : Springer-Verlag New York. - 1072-3374 .- 1573-8795. ; 202:2, s. 130-153
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Tidskriftsartikel (refereegranskat)abstract
- We construct and analyze a finite volume scheme for numerical solution of a three-dimensional Poisson equation. We derive optimal convergence rates in the discrete H1 norm and sub-optimal convergence in the maximum norm, where we use the maximal available regularity of the exact solution and minimal smoothness requirement on the source term. The theoretical results are justified through implementing some canonical examples in 3D.
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| 3. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem
- 2022
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Ingår i: Bit Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 62, s. 1383-1416
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we deal with construction and analysis of a multiwavelet spectral element scheme for a generalized Cauchy type problem with Caputo fractional derivative. Numerical schemes for this type of problems, often suffer from the draw-back of spurious oscillations. A common remedy is to render the problem to an equivalent integral equation. For the generalized Cauchy type problem, a corresponding integral equation is of nonlinear Volterra type. In this paper we investigate wellposedness and convergence of a stabilizing multiwavelet scheme for a, one-dimensional case (in [a, b] or [0, 1]), of this problem. Based on multiwavelets, we construct an approximation procedure for the fractional integral operator that yields a linear system of equations with sparse coefficient matrix. In this setting, choosing an appropriate threshold, the number of non-zero coefficients in the system is substantially reduced. A severe obstacle in the convergence analysis is the lack of continuous derivatives in the vicinity of the inflow/ starting boundary point. We overcome this issue through separating a J (mesh)-dependent, small, neighborhood of alpha (or origin) from the interval, where we only take L-2-norm. The estimate in this part relies on Chebyshev polynomials, viz. As reported by Richardson(Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms, Oxford University, UK, 2012) and decreases, almost, exponentially by raising J. At the remaining part of the domain the solution is sufficiently regular to derive the desired optimal error bound. We construct such a modified scheme and analyze its wellposedness, efficiency and accuracy. The robustness of the proposed scheme is confirmed implementing numerical examples.
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| 4. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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ON HP-STREAMLINE DIFFUSION AND NITSCHE SCHEMES FOR THE RELATIVISTIC VLASOV-MAXWELL SYSTEM
- 2019
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Ingår i: Kinetic and Related Models. - : American Institute of Mathematical Sciences (AIMS). - 1937-5093 .- 1937-5077. ; 12:1, s. 105-131
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Tidskriftsartikel (refereegranskat)abstract
- We study stability and convergence of hp-streamline diffusion (SD) finite element, and Nitsche's schemes for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system and Maxwell's equations, respectively. For the hp scheme for the Vlasov-Maxwell system, assuming that the exact solution is in the Sobolev space HS+1(Omega), we derive global a priori error bound of order O(h/p)(s+1/2), where h(= max(K) h(K)) is the mesh parameter and p(= max(K) p(K)) is the spectral order. This estimate is based on the local version with h(K) = diam K being the diameter of the phase-space-time element K and pR-is the spectral order (the degree of approximating finite element polynomial) for K. As for the Nitsche's scheme, by a simple calculus of the field equations, first we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of O(h(2) +k(2)), where h is the spatial mesh size and k is the time step. Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justification of the results, in lower dimensions, is presented and is also the subject of a forthcoming computational work [22].
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| 5. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A combined discontinuous Galerkin and finite volume scheme for multi-dimensional VPFP system
- 2011
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Ingår i: AIP Conference Proceedings. 27th International Symposium on Rarefied Gas Dynamics, RGD27; Pacific Grove, CA; United States; 10 July 2011 through 15 July 2011. - : AIP. - 0094-243X .- 1551-7616. - 9780735408890 ; 1333:Part 1, s. 57-62, s. 57-62
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Konferensbidrag (refereegranskat)abstract
- We construct a numerical scheme for the multi-dimensional Vlasov-Poisson-Fokker-Planck system based on a combined finite volume (FV) method for the Poisson equation in spatial domain and the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element in time, phase-space variables for the Vlasov-Fokker-Planck equation.
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| 6. |
- 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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| 7. |
- 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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| 8. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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On the convergence of a linearly implicit finite element method for the nonlinear Schrödinger equation
- 2024
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Ingår i: STUDIES IN APPLIED MATHEMATICS. - 0022-2526 .- 1467-9590.
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Tidskriftsartikel (refereegranskat)abstract
- We consider a model initial- and Dirichlet boundary-value problem for a nonlinear Schr & ouml;dinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second-order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal-order error estimates in the L2$L<^>2$ and the H1$H<^>1$ norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a courant-friedrichs-lewy (CFL) condition between the space mesh size and the time step sizes.
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| 9. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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Convergence of stabilized p1 finite element scheme for time harmonic maxwell’s equations
- 2020
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer International Publishing. - 2194-1017 .- 2194-1009. ; 328, s. 33-43
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Konferensbidrag (refereegranskat)abstract
- The paper considers the convergence study of the stabilized P1 finite element method for the time harmonic Maxwell’s equations. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. For the stabilized model a coercivity relation is derived that guarantee’s the existence of a unique solution for the discrete problem. The convergence is addressed both in a priori and a posteriori settings. Our numerical examples validate obtained convergence results.
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| 10. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A Discontinuous Galerkin Approach for Stabilized Maxwell’s Equations in Pseudo-Frequency Domain
- 2023
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Ingår i: Springer Proceedings in Mathematics and Statistics. - 2194-1009 .- 2194-1017. - 9783031358708
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Konferensbidrag (refereegranskat)abstract
- This paper concerns the study of a stabilized discontinuous Galerkin finite element method for the Maxwell’s equations in pseudo-frequency domain obtained through Laplace transformation in time. The model problem is considered in the special case assuming constant dielectric permittivity function in a boundary neighborhood. The discontinuous Galerkin finite element method (DGFEM) is formulated and the convergence is addressed in a priori setting where we derive optimal order error bound of the scheme in a L2 -based triple norm. Finally, our numerical examples confirm predicted convergence of the proposed scheme.
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