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![Example 5.3: discretization errors in different norms using DDC and NSP. a‖·‖ε,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _{\varepsilon ,k}$$\end{document} and H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} norms, bL∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document} and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} norms](publication/321136966/figure/fig2/AS:961816914038800@1606326430482/Example53-discretization-errors-in-different-norms-using-DDC-and-NSP.gif)
Example 5.3: discretization errors in different norms using DDC and NSP. a‖·‖ε,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _{\varepsilon ,k}$$\end{document} and H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} norms, bL∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document} and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} norms
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![Affine map x^=FKx\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/321136966/figure/fig1/AS:961816914055179@1606326430261/Affine-map-xFKxdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![Affine map x^=FKx\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/321136966/figure/fig4/AS:961816918253571@1606326431094/Affine-map-xFKxdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection–diffusion equations. Since the discretization error in a suitable norm can be bounded by the sum of interpolation error and its var...
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Citations
... The streamline upwind Petrov-Galerkin (SUPG) method is another effective means to discrete the electron and hole current continuity equations [17], [18]. However, finding the "optimal" upwinding function for multidimensional applications is extremely challenging [19], [20]. There have been many attempts to predict the definition of upwinding function through the artificial intelligence approach [21]. ...
This article presents the construction and analysis of a hybrid numerical method for the discretization of the convection-dominated nonlinear carrier transport process in semiconductor devices. In the construction of the method, the edge streamline upwind (SU) current density model is employed to overcome the nonconvergence brought by the nonlinear property of the current continuity equation, and the mixed finite volume-finite element method (FVFEM) is utilized to effectively extend the edge SU current density model to multidimensional applications. The proposed method is more reliable and flexible than the finite box (FB) method and streamline upwind Petrov-Galerkin (SUPG) method. The performance of several popular SU current density models is compared to find the optimal one. The proposed method is validated by comparing the calculated results with those calculated using commercial software. Based on the streamline upwind-finite volume-finite element method (SU-FVFEM), the finite element method (FEM), and the domain decomposition method (DDM), an in-house parallel computing electrothermal simulator is developed for large-scale semiconductor devices. The performance of the in-house developed parallel simulator is evaluated through simulations of a p-n junction diode and a 3-D bipolar transistor. The results demonstrated that the developed parallel simulator possesses good accuracy, applicability, and scalability.
... where the field b ∈ L ∞ (0, T ; W 1,∞ (Ω)) 2 is incompressible, g ∈ L 2 (0, T ; L 2 (∂Ω)), f ∈ L 2 (0, T ; L 2 (Ω)), u ∈ L 2 (0, T ; H 1 g (Ω)), and ε ∈ [0, 1) is the diffusivity coefficient. There have been attempts at numerically solving (1a) for time-independent solutions using adaptive mesh methods in conjunction with streamline upwind Petrov-Galerkin methods (SUPG) for steady flows (see, for example, [1,2]) when ε 1. Mesh adaptation has been a strong force in tackling convection-dominated problems. The general class of problems undertaken in this area of research should have the following two properties: artificial (non-physical) oscillations and sharp layers (i.e., interior and boundary layers). ...
... To improve stability, it was encouraged in [6] that artificial diffusion in the direction of b should be added to the standard Galerkin method. The SUPG method has been relatively useful in numerically solving these types of boundary/interior layer problems for time-dependent and time-independent problems [2,7,8,9]. ...
... where σ = −1, 0, 1 is the Galerkin least-squares operator, streamline upwind operator, and the multiscale operator, respectively. In our work, we use linear basis functions, hence all operators are equivalent to (2). The operators in (3) and (4) rely on using more complex basis functions than linear ones (see, for example, [10,11]). ...
We investigate the effect of the streamline upwind Petrov-Galerkin method (SUPG) as it relates to the moving mesh partial differential equation (MMPDE) method for convection-diffusion problems in the presence of vanishing diffusivity. We first discretize in space using linear finite elements and then use a $\theta$-scheme to discretize in time. On a fixed mesh, SUPG (FM-SUPG) is shown to enhance the stability and resolves spurious oscillations when compared to the classic Galerkin method (FM-FEM) when diffusivity is small. However, it falls short when the layer-gradient is large. In this paper, we develop a moving mesh upwind Petrov-Galerkin (MM-SUPG) method by integrating the SUPG method with the MMPDE method. Numerical results show that our MM-SUPG works well for these types of problems and performs better than FM-SUPG as well as MMPDE without SUPG.
