Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems
Chongqing Jiaotong University（重庆交通大学）；Texas A&M International University（德州农工国际大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Wayne State University（韦恩州立大学）
In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for singularly perturbed convection-diffusion-reaction equations in a new dual norm presented in [Du and Zhang, J. Sci. Comput. (2015)]. The flux is recovered by either local averaging in conforming $H({\rm div})$ spaces or weighted global $L^2$ projection onto conforming $H({\rm div})$ spaces. We further introduce a recovery stabilization procedure, and develop completely robust a posteriori error estimators with respect to the singular perturbation parameter $\varepsilon$. Numerical experiments are reported to support the theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in $\varepsilon$.
Upwinded mass fluxes are described and analysed for advection operators discretised using mixed mimetic spectral elements. This involves a Petrov-Galerkin formulation by which the mass flux test functions are evaluated at downstream locations along velocity characteristics. As for the original mixed mimetic spectral element advection operator, the upwinded mass flux advection operator is conservative, however unlike the original advection operator, which is purely hyperbolic, the upwinded advection operator adds dissipation which is biased towards high wave numbers. The upwinded advection oper…
The present study deals with the finite element discretization of nanofluid convective transport in an enclosure with variable properties. We study the Buongiorno model, which couples the Navier-Stokes equations for the base fluid, an advective-diffusion equation for the heat transfer, and an advection dominated nanoparticle fraction concentration subject to thermophoresis and Brownian motion forces. We develop an iterative numerical scheme that combines Newton's method (dedicated to the resolution of the momentum and energy equations) with the transport equation that governs the nanoparticles…
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are pre…
A two-parameter singularly perturbed problem with discontinuous source and convection coefficient is considered in one dimension. Both convection coefficient and source term are discontinuous at a point in the domain. The presence of perturbation parameters results in boundary layers at the boundaries. Also, an interior layer occurs due to the discontinuity of data at an interior point. An upwind scheme on an appropriately defined Shishkin-Bakhvalov mesh is used to resolve the boundary layers and interior layers. A three-point formula is used at the point of discontinuity. The proposed method …
In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow exponential boundary layers. Parameter-explicit theoretical bounds on the derivatives of the analytical solution are given, which are used in the error analysis of the proposed scheme. The problem is discretized by a hybrid finite difference scheme comprises of midpoint-upwind and central difference operator on an appropriate piecewise-uniform fitted mesh. …
In this article, we have considered a time-dependent two-parameter singularly perturbed parabolic problem with discontinuous convection coefficient and source term. The problem contains the parameters $ε$ and $μ$ multiplying the diffusion and convection coefficients, respectively. A boundary layer develops on both sides of the boundaries as a result of these parameters. An interior layer forms near the point of discontinuity due to the discontinuity in the convection and source term. The width of the interior and boundary layers depends on the ratio of the perturbation parameters. We discuss t…
We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-upwind scheme proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), pp. 133--160]. The designed scheme is well-balanced in the sense that it is capable of exactly preserving "lake-at-rest" steady states. In order to achieve this as well as to preserve positivity of water…
Conductor moving in magnetic field is quite common in electrical equipment. The numerical simulation of such problem is vital in their design and analysis of electrical equipment. The Galerkin finite element method (GFEM) is a commonly employed simulation tool, nonetheless, due to its inherent numerical instability at higher velocities, the GFEM requires upwinding techniques to handle moving conductor problems. The Streamline Upwinding/Petrov-Galerkin (SU/PG) scheme is a widely acknowledged upwinding technique, despite its error-peaking at the transverse boundary. This error at the transverse-…
Galerkin and Petrov-Galerkin projection-based reduced-order models (ROMs) of transient partial differential equations are typically obtained by performing a dimension reduction and projection process that is defined at either the spatially continuous or spatially discrete level. In both cases, it is common to add stabilization to the resulting ROM to increase the stability and accuracy of the method; the addition of stabilization is particularly common for advection-dominated systems when the ROM is under-resolved. While these two approaches can be equivalent in certain settings, differing tec…
Modeling and Stabilization of Transport-Dominated Flows
NJIT（新泽西理工学院）；Western University（韦仕敦大学）；Purdue University（普渡大学）；U. of Houston（休斯顿大学）；NC State University（北卡罗来纳州立大学）；U. of Arizona（亚利桑那大学）；U. of Pittsburgh（匹兹堡大学）；U. of Tennessee（田纳西大学）；San Diego State University（圣地亚哥州立大学）
