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Removing the cell resonance error in the multiscale finite element method via a PetrovGalerkin formulation
 Communications in Mathematical Sciences
, 2004
"... Abstract. We continue the study of the nonconforming multiscale finite element method (MsFEM) introduced in [17, 14] for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. I ..."
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Abstract. We continue the study of the nonconforming multiscale finite element method (MsFEM) introduced in [17, 14] for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an oversampling technique. Nonetheless, there is still a secondary cell resonance error of O(ε 2 /h 2). Here, we introduce a PetrovGalerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersedinterface finite element method for elliptic interface problems.
A Summary of Numerical Methods for TimeDependent AdvectionDominated Partial Differential Equations
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 2000
"... We give a brief summary of numerical methods for timedependent advectiondominated partial differential equations (PDEs), including firstorder hyperbolic PDEs and nonstationary advectiondiffusion PDEs. Mathematical models arising in porous medium fluid flow are presented to motivate these equatio ..."
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Cited by 14 (0 self)
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We give a brief summary of numerical methods for timedependent advectiondominated partial differential equations (PDEs), including firstorder hyperbolic PDEs and nonstationary advectiondiffusion PDEs. Mathematical models arising in porous medium fluid flow are presented to motivate these equations. It is understood that these PDEs also arise in many other important fields and that the numerical methods reviewed apply to general advectiondominated PDEs. We conduct a brief historical review of classical numerical methods, and a survey of the recent developments on the Eulerian and characteristic methods for timedependent advectiondominated PDEs. The survey is not comprehensive due to the limitation of its length, and a large portion of the paper covers characteristic or EulerianLagrangian methods.
Unstructured Grid Finite Element Methods for Fluid Mechanics
, 1997
"... . The development of unstructured grid based finite element methods for the simulation of fluid flows is reviewed. The review concentrates on solution techniques for the compressible Euler and Navier Stokes equations, employing methods which are based upon a Galerkin discretisation in space together ..."
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Cited by 13 (4 self)
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. The development of unstructured grid based finite element methods for the simulation of fluid flows is reviewed. The review concentrates on solution techniques for the compressible Euler and Navier Stokes equations, employing methods which are based upon a Galerkin discretisation in space together with an appropriate finite difference representation in time. It is assumed that unstructured assemblies of triangles are used to achieve the spatial discretisation in two dimensions, with unstructured assemblies of tetrahedra employed in the three dimensional case. Adaptive grid procedures are discussed and methods for accelerating the iterative solution convergence are considered. The areas of incompressible flow modelling and optimisation are also included. 1. Introduction Over the past thirty years, there has been an intense research activity in the area of computational fluid dynamics. A large proportion of this activity has been driven by the aerospace industry, with its requirements...
Eigenvalue solution of thermoelastic instability problems using Fourier reduction
, 2000
"... A finiteelement method is developed for determining the critical sliding speed for thermoelastic instability of an axisymmetric clutch or brake. Linear perturbations on the constantspeed solution are sought that vary sinusoidally in the circumferential direction and grow exponentially in time. The ..."
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Cited by 10 (3 self)
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A finiteelement method is developed for determining the critical sliding speed for thermoelastic instability of an axisymmetric clutch or brake. Linear perturbations on the constantspeed solution are sought that vary sinusoidally in the circumferential direction and grow exponentially in time. These factors cancel in the governing thermoelastic and heatconduction equations, leading to a linear eigenvalue problem on the twodimensional crosssectional domain for the exponential growth rate for each Fourier wavenumber. The imaginary part of this growth rate corresponds to a migration of the perturbation in the circumferential direction. The algorithm is tested against an analytical solution for a layer sliding between two halfplanes and gives excellent agreement, for both the critical speed and the migration speed. Criteria are developed to determine the mesh refinement required to give an adequate discrete description of the thermal boundary layer adjacent to the sliding interface. The method is then used to determine the unstable mode and critical speed in geometries approximating current multidisc clutch practice.
An hp Finite Element Method for convectiondiffusion problems
, 1997
"... We analyze an hp FEM for convectiondiffusion problems. Stability is achieved by suitably upwinded test functions, generalizing the classical ffquadratically upwinded and the Hemker testfunctions for piecewise linear trial spaces (see, e.g., [12] and the references there). The method is proved to ..."
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Cited by 8 (1 self)
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We analyze an hp FEM for convectiondiffusion problems. Stability is achieved by suitably upwinded test functions, generalizing the classical ffquadratically upwinded and the Hemker testfunctions for piecewise linear trial spaces (see, e.g., [12] and the references there). The method is proved to be stable independently of the viscosity. Further, the stability is shown to depend only weakly on the spectral order. We show how sufficiently accurate, approximate upwinded test functions can be computed on each element by a local least squares FEM. Under the assumption of analyticity of the input data, we prove robust exponential convergence of the method. Numerical experiments confirm our convergence estimates and show robust exponential convergence of the hpFEM even for viscosities of the order of machine precision, i.e., for the limiting transport problem.
An Ellam Scheme For MultiDimensional AdvectionReaction Equations And Its OptimalOrder Error Estimate
 SIAM J. Numer. Anal
, 2001
"... . We present an ELLAM (EulerianLagrangian localized adjoint method) scheme for initialboundary value problems for advectionreaction partial dierential equations in multiple space dimensions. The derived numerical scheme is not subject to the CFL (CourantFriedrichsLewy) condition and generates a ..."
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. We present an ELLAM (EulerianLagrangian localized adjoint method) scheme for initialboundary value problems for advectionreaction partial dierential equations in multiple space dimensions. The derived numerical scheme is not subject to the CFL (CourantFriedrichsLewy) condition and generates accurate numerical solutions even if large time steps are used. Moreover, the scheme naturally incorporates boundary conditions into its formulation without any articial outow boundary conditions needed, and conserves mass. An optimalorder error estimate is proved for the scheme. Numerical experiments are performed to verify the theoretical estimate. Key words. characteristic methods, convergence analysis, error estimates, EulerianLagrangian methods, numerical simulation of advectionreaction equations AMS subject classications. 65M25, 65M60, 76M10, 76S05 1. Introduction. Advectiondominated reactive transport partial dierential equations (PDEs) arise in petroleum reservoir simulatio...
SecondOrder Characteristic Methods for AdvectionDiffusion Equations and Comparison to Other Schemes
 Advances in Water Resources
, 1999
"... We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order RungeKutta approximation of the characteristics within the framework of the EulerianLagrangian localized adjoint method. These methods naturally incorporate all three type ..."
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Cited by 6 (2 self)
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We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order RungeKutta approximation of the characteristics within the framework of the EulerianLagrangian localized adjoint method. These methods naturally incorporate all three types of boundary conditions in their formulations, are fully mass conservative, and generate regularly structured systems which are symmetric and positive definite for most combinations of the boundary conditions. Extensive numerical experiments are presented which compare the performance of these two RungeKutta methods to many other well perceived and widely used methods which include many Galerkin methods and high resolution methods from #uid dynamics. Key words characteristic methods, comparison of numerical methods, EulerianLagrangian methods, numerical solutions of advectiondi#usion equations, RungeKutta methods. 1 Introduction Advectiondi#usion equations are an important cla...
A combination of time domain finite elementboundari integral and with time domain physical optics for calculation of electromagnetic scattering of 3D structures
 Progress In Electromagnetics Research, PIER 79, 463–474
, 2008
"... Abstract—This paper presents a hybrid numerical approach combining an improved Time Domain Finite ElementBoundary Integral (FEBI) method with Time Domain Physical Optics (TDPO) for calculations of electromagnetic scattering of 3D combinativecomplex objects. For complexcombined objects containin ..."
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Abstract—This paper presents a hybrid numerical approach combining an improved Time Domain Finite ElementBoundary Integral (FEBI) method with Time Domain Physical Optics (TDPO) for calculations of electromagnetic scattering of 3D combinativecomplex objects. For complexcombined objects containing a small size and large size parts, using TDPO is an appropriate approach for coupling between two regions. Therefore, our technique calculates the objects complexity with the help of FEBI and the combinatory structures by using of the TDPO. The hybridization algorithm for restrictive object is implemented and the numerical results validate the superiority of the proposed algorithm via realistic electromagnetic applications. 1.
Stability and convergence of a finite element method for reactive transport in ground water
 SIAM J. Numer. Anal
, 1997
"... Abstract. An explicit nite element method is used to solve the linear convectiondi usionreaction equations governing contaminant transport in ground water owing through an adsorbing porous medium. The use of discontinuous nite elements for the convective part of the equations combined with mixed ni ..."
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Cited by 6 (5 self)
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Abstract. An explicit nite element method is used to solve the linear convectiondi usionreaction equations governing contaminant transport in ground water owing through an adsorbing porous medium. The use of discontinuous nite elements for the convective part of the equations combined with mixed nite elements for the di usive part renders the method for the concentration solution, which displays strong gradients, trivially conservative and fully parallelizable. We carry out a stability and convergence analysis. In particular, the method is proven to satisfy a maximum principle, to be total variation bounded, and to converge to the unique weak solution of the equations. Special attention is paid to the convective part of the equations. Numerical simulations are presented and discussed. nite element and volume method, conKey words. convectiondi usionreaction equation, servation law, stability, convergence, mixed method
Modified Streamline Diffusion Schemes For ConvectionDiffusion Problems
, 1997
"... . We consider the design of robust and accurate finite element approximation methods for solving convectiondiffusion problems. We develop some twoparameter streamline diffusion schemes with piecewise bilinear (or linear) trial functions and show that these schemes satisfy the necessary condition ..."
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Cited by 5 (1 self)
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. We consider the design of robust and accurate finite element approximation methods for solving convectiondiffusion problems. We develop some twoparameter streamline diffusion schemes with piecewise bilinear (or linear) trial functions and show that these schemes satisfy the necessary conditions for L 2 uniform convergence of order greater than 1=2 introduced by Stynes and Tobiska. For smooth problems, the schemes satisfy error bounds of the form O(h)juj2 in an energy norm. In addition, extensive numerical experiments show that they effectively reproduce boundary layers and internal layers caused by discontinuities on relatively coarse grids, without any requirements on alignment of flow and grid. Key words. Convectiondiffusion, streamline diffusion, crosswind diffusion, boundary layer, characteristic layer. AMS(MOS) subject classifications. primary 65N30, 65F10 1. Introduction. Consider the twodimensional convectiondiffusion equation \Gamma"\Deltau + fi \Delta ru = f ...