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| 2. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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A Discontinuous Galerkin Approach for Stabilized Maxwell’s Equations in Pseudo-Frequency Domain
- 2023
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In: Springer Proceedings in Mathematics and Statistics. - 2194-1009 .- 2194-1017. - 9783031358708
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Conference paper (peer-reviewed)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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| 3. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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A posteriori error estimates in a globally convergent FEM for a hyperbolic coefficient inverse problem
- 2010
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In: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 26:11
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Journal article (peer-reviewed)abstract
- This study concerns a posteriori error estimates in a globally convergent numerical method for a hyperbolic coefficient inverse problem. Using the Laplace transform the model problem is reduced to a nonlinear elliptic equation with a gradient dependent nonlinearity. We investigate the behavior of the nonlinear term in both a priori and a posteriori settings and derive optimal a posteriori error estimates for a finite-element approximation of this problem. Numerical experiments justify the efficiency of a posteriori estimates in the globally convergent approach.
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| 4. |
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| 5. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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A Priori Error Estimates and Computational Studies for a Fermi Pencil-Beam Equation
- 2018
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In: Journal of Computational and Theoretical Transport. - : Informa UK Limited. - 2332-4325 .- 2332-4309. ; 47:1-3, s. 125-151
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Journal article (peer-reviewed)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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| 6. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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A stabilized P1 domain decomposition finite element method for time harmonic Maxwell's equations
- 2023
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In: Mathematics and Computers in Simulation. - : Elsevier BV. - 0378-4754. ; 204, s. 556-574
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Journal article (peer-reviewed)abstract
- One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell's equations is to render their hyperbolic character to elliptic form. This paper is devoted to the study of a stabilized linear, domain decomposition, finite element method for the time harmonic Maxwell's equations, in a dual form, obtained through the Laplace transformation in time. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. The discrete problem is coercive in a symmetrized norm, equivalent to the discrete norm of the model problem. This yields discrete stability, which together with continuity guarantees the well-posedness of the discrete problem, cf Arnold et al. (2002) [3], Di Pietro and Ern (2012) [45]. The convergence is addressed both in a priori and a posteriori settings. In the a priori error estimates we confirm the theoretical convergence of the scheme in a L2-based, gradient dependent, triple norm. The order of convergence is O(h) in weighted Sobolev space Hw2(ohm), and hence optimal. Here, the weight w := w(epsilon, s) where epsilon is the dielectric permittivity function and s is the Laplace transformation variable. We also derive, similar, optimal a posteriori error estimates controlled by a certain, weighted, norm of the residuals of the computed solution over the domain and at the boundary (involving the relevant jump terns) and hence independent of the unknown exact solution. The a posteriori approach is used, e.g. in constructing adaptive algorithms for the computational purposes, which is the subject of a forthcoming paper. Finally, through implementing several numerical examples, we validate the robustness of the proposed scheme. (c) 2022 The Author(s). Published by Elsevier B.V. on behalf of International Association for Mathematics and Computers in Simulation (IMACS). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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| 8. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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Adaptive approximate globally convergent algorithm with backscattered data.
- 2013
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In: Inverse Problems and Large-Scale Computations. Springer Proceedings in Mathematics and Statistics. Larisa Beilina, Yury V. Shestopalov (Eds.). - Cham : Springer International Publishing. - 2194-1009 .- 2194-1017. - 9783319006598 ; 52, s. 1-20
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Conference paper (peer-reviewed)abstract
- We construct, analyze and implement an approximately globally convergent finite element scheme for a hyperbolic coefficient inverse problem in the case of backscattering data. This extends the computational aspects introduced in Asadzadeh and Beilina (Inv. Probl. 26, 115007, 2010), where using Laplace transformation, the continuous problem is reduced to a nonlinear elliptic equation with a gradient dependent nonlinearity. We investigate the behavior of the nonlinear term and discuss the stability issues as well as optimal a posteriori error bounds, based on an adaptive procedure and due to the maximal available regularity of the exact solution. Numerical implementations justify the efficiency of adaptive a posteriori approach in the globally convergent setting.
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| 9. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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Finite element schemes for Fermi equation
- 2017
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In: International Conference of Numerical Analysis and Applied Mathematics 2016, ICNAAM 2016, Rhodes, Greece, 19-25 September 2016. AIP Conference Proceedings. Vol. 1863, nr. 1, 370007. - : Author(s). - 0094-243X .- 1551-7616. - 9780735415386
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Conference paper (peer-reviewed)abstract
- A priori error estimates are derived for the streamline diffusion (SD) finite element methods for the Fermi pencil-beam equation. Two-dimensional numerical examples confirm our theoretical investigations.
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| 10. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
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Stability and Convergence Analysis of a Domain Decomposition FE/FD Method for Maxwell's Equations in the Time Domain
- 2022
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In: Algorithms. - : MDPI AG. - 1999-4893. ; 15:10
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Journal article (peer-reviewed)abstract
- Stability and convergence analyses for the domain decomposition finite element/finite difference (FE/FD) method are presented. The analyses are developed for a semi-discrete finite element scheme for time-dependent Maxwell's equations. The explicit finite element schemes in different settings of the spatial domain are constructed and a domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.
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