In order to obtain an expected numerical solution, a fully discrete discontinuous Galerkin method is applied to a kind of reactive transport problems in two dimension. That is to say, the space variable is discretized...In order to obtain an expected numerical solution, a fully discrete discontinuous Galerkin method is applied to a kind of reactive transport problems in two dimension. That is to say, the space variable is discretized with the symmetric interior penalty Calerkin method （SIPG）, and the time variable is done with the backward Euler method. Making use of the duality technique, hp approximation properties and the interpolation theory, a residual-type a posteriori error estimation is achieved, which can be used for adaptivity. Compared with the analyses of semi-discretization, the current presentation is more challenging and more significant.展开更多
In this paper we review a series of recent work on using a Fourier analysis technique to study the sta-bility and error estimates for the discontinuous Galerkin method and other related schemes. The ad-vantage of this...In this paper we review a series of recent work on using a Fourier analysis technique to study the sta-bility and error estimates for the discontinuous Galerkin method and other related schemes. The ad-vantage of this approach is that it can reveal instability of certain "bad" ' schemes; it can verify stability for certain good schemes which are not easily amendable to standard finite element stability analysis techniques; it can provide quantitative error comparisons among different schemes; and it can be used to study superconvergence and time evolution of errors for the discontinuous Galerkin method. We will briefly describe this Fourier analysis technique, summarize its usage in stability and error estimates for various schemes, and indicate the advantages and disadvantages of this technique in comparison with other finite element techniques.展开更多
This paper applies bilinear immersed finite elements （IFEs） in the interior penalty discontinuous Galerkin （DG） methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous ...This paper applies bilinear immersed finite elements （IFEs） in the interior penalty discontinuous Galerkin （DG） methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous bihnear IFE space is constructed and applied to both the symmetric and nonsymmetric interior penalty DG formulations. The new methods can solve an interface problem on a Cartesian mesh independent of the interface with local refinement at any locations needed even if the interface has a nontrivial geometry. Numerical examples are provided to show features of these methods.展开更多
基金supported by Hunan Provincial Natural Science Foundation of China under Grant No. 10JJ3021Scientific Research Fund of Hunan Provincial Education Department under Grant No.11B032the Planned Science and Technology Project of Hunan Province and Aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province
文摘In order to obtain an expected numerical solution, a fully discrete discontinuous Galerkin method is applied to a kind of reactive transport problems in two dimension. That is to say, the space variable is discretized with the symmetric interior penalty Calerkin method （SIPG）, and the time variable is done with the backward Euler method. Making use of the duality technique, hp approximation properties and the interpolation theory, a residual-type a posteriori error estimation is achieved, which can be used for adaptivity. Compared with the analyses of semi-discretization, the current presentation is more challenging and more significant.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10671190, 10671190)NSF Grant DMS-0809086DOE Grant DE-FG02-08ER25863
文摘In this paper we review a series of recent work on using a Fourier analysis technique to study the sta-bility and error estimates for the discontinuous Galerkin method and other related schemes. The ad-vantage of this approach is that it can reveal instability of certain "bad" ' schemes; it can verify stability for certain good schemes which are not easily amendable to standard finite element stability analysis techniques; it can provide quantitative error comparisons among different schemes; and it can be used to study superconvergence and time evolution of errors for the discontinuous Galerkin method. We will briefly describe this Fourier analysis technique, summarize its usage in stability and error estimates for various schemes, and indicate the advantages and disadvantages of this technique in comparison with other finite element techniques.
基金Supported by the National Natural Science Fund (11061021)the National Science Foundation of Inner Mongolia (2012MS0106)the Endancing Comprehensive Strength Project of Inner Mongolia University (14020202)
基金supported by NSF grant DMS-0713763the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 501709)the AMSS-PolyU Joint Research Institute for Engineering and Management Mathematics, and NSERC (Canada)
文摘This paper applies bilinear immersed finite elements （IFEs） in the interior penalty discontinuous Galerkin （DG） methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous bihnear IFE space is constructed and applied to both the symmetric and nonsymmetric interior penalty DG formulations. The new methods can solve an interface problem on a Cartesian mesh independent of the interface with local refinement at any locations needed even if the interface has a nontrivial geometry. Numerical examples are provided to show features of these methods.
