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

2015
*
Discrete Dynamics in Nature and Society
*

We present

doi:10.1155/2015/985214
fatcat:ldlsvnyajjfenjxm7mbqe3ukuy
*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. ...##
###
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

2015
*
Discrete Dynamics in Nature and Society
*

We combine theH1-

doi:10.1155/2015/618258
fatcat:hbn2uh4djza2popgjqquflcue4
*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. ...##
###
A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method

1983
*
Mathematics of Computation
*

We consider

doi:10.1090/s0025-5718-1983-0717692-8
fatcat:jrblr4lq7jggfnsvz4vydcgpxa
*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*. ...##
###
A Spline-Trigonometric Galerkin Method and an Exponentially Convergent Boundary Integral Method

1983
*
Mathematics of Computation
*

We consider

doi:10.2307/2007682
fatcat:xuj54bqigvakzbleymsjh55pdm
*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*. ...##
###
Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable

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. ...

##
###
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

2019
*
Communications on Applied Mathematics and Computation
*

By

doi:10.1007/s42967-019-00037-6
fatcat:rgm4zk5ajjaqta3jzxmtovw3pe
*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. ...##
###
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

1990
*
Bulletin of the American Mathematical Society
*

It is certainly more convenient than

doi:10.1090/s0273-0979-1990-16009-4
fatcat:goeklfrjuvdgro6hkhseods2yu
*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*. ...##
###
Development of Galerkin Method for Solving the Generalized Burger's-Huxley Equation

2013
*
Mathematical Problems in Engineering
*

Numerical treatments

doi:10.1155/2013/165492
fatcat:qp6rilztu5hmvliy3alvsrr2yu
*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*. ...##
###
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

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. ...

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

1999
*
Computers and Mathematics with Applications
*

We investigate

doi:10.1016/s0898-1221(99)00264-3
fatcat:ptzhymh7erenphixo5aml73s7i
*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*. ...##
###
Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

2013
*
International Journal of Computer Mathematics
*

Discontinuous

doi:10.1080/00207160.2013.784280
fatcat:o6u4wqupgnee3a52pgkwbirm3m
*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). ...
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