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A Time Discontinuous Galerkin Finite Element Method for Quasi-Linear Sobolev Equations

Hong Yu, Tongjun Sun
2015 Discrete Dynamics in Nature and Society  
We present a time discontinuous Galerkin finite element scheme for quasi-linear Sobolev equations.  ...  The approximate solution is sought as a piecewise polynomial of degree in time variable at mostq-1with coefficients in finite element space.  ...  Acknowledgments The second author is supported by the NSF of China (nos. 11271231 and 11301300). The authors would like to thank the anonymous referees for their comments and suggestions on the paper.  ... 
doi:10.1155/2015/985214 fatcat:ldlsvnyajjfenjxm7mbqe3ukuy

Page 6427 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews  
The authors consider continuous-in-time mixed finite element methods for linear Sobolev equations. They prove optimal or- der convergence results in L*(L*) norms for associated variables.  ...  Summary: “Space-time discontinuous Galerkin (DG) methods provide a solution for elastodynamic analysis, a problem that serves as a model for DG approximation of second-order hyper- bolic problems.  ... 

The Time DiscontinuousH1-Galerkin Mixed Finite Element Method for Linear Sobolev Equations

Hong Yu, Tongjun Sun, Na Li
2015 Discrete Dynamics in Nature and Society  
We combine theH1-Galerkin mixed finite element method with the time discontinuous Galerkin method to approximate linear Sobolev equations. The advantages of these two methods are fully utilized.  ...  The existence and uniqueness of the solutions are proved, and the optimalH1-norm error estimates are derived. We get high accuracy for both the space and time variables.  ...  Acknowledgments The second author is supported by the NSF of China (nos. 11271231 and 11301300). The authors would like to thank the anonymous referees for their comments and suggestions on the paper.  ... 
doi:10.1155/2015/618258 fatcat:hbn2uh4djza2popgjqquflcue4

A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method

Douglas N. Arnold
1983 Mathematics of Computation  
We consider a Galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions.  ...  In constrast to ordinary spline Galerkin methods, we show that the method is stable, and so provides quasioptimal approximation, in a large family of Hubert spaces including all the Sobolev spaces of negative  ...  We shall study the spline-trig method for the integral equation (2. 3). This is a Galerkin method employing different test and trial spaces.  ... 
doi:10.1090/s0025-5718-1983-0717692-8 fatcat:jrblr4lq7jggfnsvz4vydcgpxa

A Spline-Trigonometric Galerkin Method and an Exponentially Convergent Boundary Integral Method

Douglas N. Arnold
1983 Mathematics of Computation  
We consider a Galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions.  ...  In constrast to ordinary spline Galerkin methods, we show that the method is stable, and so provides quasioptimal approximation, in a large family of Hubert spaces including all the Sobolev spaces of negative  ...  We shall study the spline-trig method for the integral equation (2. 3). This is a Galerkin method employing different test and trial spaces.  ... 
doi:10.2307/2007682 fatcat:xuj54bqigvakzbleymsjh55pdm

Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable

Douglas N. Arnold, Jim Douglas, Vidar Thom{ée
1981 Mathematics of Computation  
A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed.  ...  Discretization in time by explicit single-step methods is discussed.  ...  The Eqs. (2.2a) can be interpreted as a finite system of ordinary differential equations in the coefficients of U with respect to some basis for 911.  ... 
doi:10.1090/s0025-5718-1981-0595041-4 fatcat:4mcjxacosrgtthpxa3mahxofba

Page 1333 of Mathematical Reviews Vol. 51, Issue 4 [page]

1976 Mathematical Reviews  
The use of Richardson extrapolation in conjunction with several discrete-time Galerkin methods for the approximate solution of parabolic initial-boundary value problems is investigated.  ...  From the authors’ summary: “Interior a priori error estimates in Sobolev norms are derived from interior Ritz-Galerkin equations which are common to a class of methods used in approximating solutions of  ... 

An Indirect Finite Element Method for Variable-Coefficient Space-Fractional Diffusion Equations and Its Optimal-Order Error Estimates

Xiangcheng Zheng, V. J. Ervin, Hong Wang
2019 Communications on Applied Mathematics and Computation  
By the representation formula of the solutions u(x) to the proposed variable coefficient models in terms of v(x), the solutions to the constant coefficient analogues, we apply finite element methods for  ...  We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension.  ...  A Petrov-Galerkin weak formulation for variable-coefficient sFDEs.  ... 
doi:10.1007/s42967-019-00037-6 fatcat:rgm4zk5ajjaqta3jzxmtovw3pe

Page 7782 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
The approach is based on a simultaneous discretization of space and time variables by using continuous finite element methods.  ...  Summary: “Galerkin and weighted Galerkin methods are pro- posed for the numerical solution of parabolic partial differential equations where the diffusion coefficient takes different signs.  ... 

Book Review: An introduction to the numerical analysis of spectral methods

Levi Lustman
1990 Bulletin of the American Mathematical Society  
It is certainly more convenient than the other methods: to use the Galerkin method for a term a(x)u{x) one must find the coefficients of au from the coefficients of u and the multiplier a, while collocation  ...  A different situation occurs when the function to be expanded possesses a few continuous derivatives but is not in C°°-a natural setting for such functions is Sobolev space.  ... 
doi:10.1090/s0273-0979-1990-16009-4 fatcat:goeklfrjuvdgro6hkhseods2yu

Development of Galerkin Method for Solving the Generalized Burger's-Huxley Equation

M. El-Kady, S. M. El-Sayed, H. E. Fathy
2013 Mathematical Problems in Engineering  
Numerical treatments for the generalized Burger's—Huxley GBH equation are presented. The treatments are based on cardinal Chebyshev and Legendre basis functions with Galerkin method.  ...  Gauss quadrature formula and El-gendi method are used to convert the problem into a system of ordinary differential equations.  ...  In this study, the spectral collocation methods with the fourth-order Runge-Kutta method for time integration are used to solve the GBH equation.  ... 
doi:10.1155/2013/165492 fatcat:qp6rilztu5hmvliy3alvsrr2yu

Page 7348 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
“Problem (1)-(3) has been studied using various methods. A. N. Botsenyuk established the existence of a generalized solution in a Hilbert space and its regularity with respect to the variable t.  ...  The Cauchy problem for the Sobolev-type equation Lu’(t) = Mu(t) (t > 0), u(0) = up is studied, where U and F are Ba- nach spaces, L is a continuous linear operator and M:dom(M) — F is a closed linear operator  ... 

A splitting mixed space-time discontinuous Galerkin method for parabolic problems

Siriguleng He, Hong Li, Yang Liu, Zhichao Fang, Jingbo Yang, Xianbiao Jia
2012 Procedia Engineering  
A splitting mixed space-time discontinuous Galerkin method is formulated to solve a class of parabolic problems.  ...  The finite element approximation of the stress is solved by time discontinuous Galerkin method with high accuracy.  ...  Acknowledgements The work of the authors is supported by the National Nature Science Foundation of China (grant no. 11061021 innovative talents training program of Inner Mongolia University.  ... 
doi:10.1016/j.proeng.2012.01.1141 fatcat:q7sgsgmwnfdh3oxahei2ayeoje

The numerical solution of Symm's equation on smooth open arcs by spline Galerkin methods

F.-J. Sayas
1999 Computers and Mathematics with Applications  
We investigate the existence of an asymptotic expansion for the error of the Galerkin method with splines on a uniform mesh as test-trial functions.  ...  Applying the well-known cosine change of variable, the arc is reparametrized and the problem is transformed into a new integral equation.  ...  AN ALTERNATIVE APPROACH We have so far studied two discretization methods for Symm's equation with the cosine change of variable in the frame of periodic even Sobolev spaces.  ... 
doi:10.1016/s0898-1221(99)00264-3 fatcat:ptzhymh7erenphixo5aml73s7i

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

Kun Liu, Béatrice M. Rivière
2013 International Journal of Computer Mathematics  
Discontinuous Galerkin Methods The discontinuous Galerkin (DG) method is a class of variational methods for solving partial differential equations by approximating the solution by discontinuous piecewise  ...  PE Now, we are ready to introduce the broken Sobolev spaces. Definition 4.1 The broken Sobolev spaces for any real number s are defined: with the broken Sobolev norm: Sobolev space HS(Ch).  ... 
doi:10.1080/00207160.2013.784280 fatcat:o6u4wqupgnee3a52pgkwbirm3m
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