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Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations [article]

Binjie Li, Tao Wang, Xiaoping Xie
2020 arXiv   pre-print
Two numerical methods with graded temporal grids are analyzed for fractional evolution equations.  ...  One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order 0<α<1, and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order  ...  Let us briefly introduce four types of numerical methods for the discretization of time fractional evolution equations.  ... 
arXiv:2002.11914v2 fatcat:otxe3rsfhbfnhjfdb4ovpgb4ui

Complex Boundary Value Problems of Nonlinear Differential Equations 2014

Xinguang Zhang, Yong Hong Wu, Dragoș-Pãtru Covei, Xinan Hao
2014 Abstract and Applied Analysis  
H and one fine grid with grid size h, respectively, the authors consider the two-grid finite element discretization techniques for the second-order nonlinear hyperbolic problems.  ...  In the paper titled "A two-grid finite element method for a second-order nonlinear hyperbolic equation, " based on two conforming piecewise linear finite element spaces on one coarse grid with grid size  ...  H and one fine grid with grid size h, respectively, the authors consider the two-grid finite element discretization techniques for the second-order nonlinear hyperbolic problems.  ... 
doi:10.1155/2014/496350 fatcat:vohr34wmsndohhpsun74csnarm

A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations [article]

Pin Lyu, Seakweng Vong
2021 arXiv   pre-print
We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations.  ...  Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for the two kinds of difference schemes by H^2 energy method.  ...  With A1-A2, (D β τ g) n−θ , g n−θ ≥ 1 2 n k=1 A (n) n−k ∇ τ ( g k 2 ) for 1 ≤ n ≤ N. (3.6) A discrete fractional Grönwall inequality proposed in [17, Theorem 3.1] is a crucial tool in the numerical analysis  ... 
arXiv:2101.09678v3 fatcat:eyidssjtejcs3a3hew2sbarthy

A linear Galerkin numerical method for a quasilinear subdiffusion equation [article]

Łukasz Płociniczak
2022 arXiv   pre-print
We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations.  ...  The temporal order depends on solution's regularity in time. Further, we support our results with numerical simulations that utilize parallelism for spatial discretization.  ...  Acknowledgement L.P. has been supported by the National Science Centre, Poland (NCN) under the grant Sonata Bis with a number NCN 2020/38/E/ST1/00153.  ... 
arXiv:2107.10057v2 fatcat:eox6khihq5e5hhcd4u3samxhrm

A time-discontinuous Galerkin method for the dynamical analysis of porous media

Zhiyun Chen, Holger Steeb, Stefan Diebels
2006 International journal for numerical and analytical methods in geomechanics (Print)  
The numerical scheme consists of finite element discretizations in the spatial and in the temporal domain simultaneously. In particular, two major classes of approaches have been investigated.  ...  This thesis deals with coupled space-time discontinuous Galerkin methods for the modeling of dynamical phenomena in fluid saturated porous media.  ...  Acknowledgment The authors are grateful to the DFG (German Science Foundation -Deutsche Forschungsgemeinschaft) for their financial support through grant number Di 430/4-1.  ... 
doi:10.1002/nag.516 fatcat:ks65x5rcvzhufc4k32edrnbesq

subject index volumes 201 to 210

2007 Journal of Computational and Applied Mathematics  
206 1082 Two-dimensional Burgers' equation 206 432 Two-dimensional systems 204 231 Two-grid method 205 509 Two-parameter dependent nonlinear problem 202 177 Two-phase flows 203 376 Two-point G 2 Hermite  ...  methods 201 164; 205 957, 777, 849 Numerical model 204 159 Numerical simulation of delay differential equations 205 835 Numerical simulations 204 18, 187 Numerical solution 205 1002; 206 174 Numerical  ... 
doi:10.1016/s0377-0427(07)00502-x fatcat:tovvilkoczetvl423gcjxug7gm

Numerical Stabilization of the Melt Front for Laser Beam Cutting [chapter]

Torsten Adolph, Willi Schönauer, Markus Niessen, Wolfgang Schulz
2010 Numerical Mathematics and Advanced Applications 2009  
Numerical experiments will be done for simple linear problems, the Burger's equation, and for the Euler equations with high Mach numbers. 24 This contribution deals with the modeling of soft biological  ...  We treat two types of boundary conditions: Dirichlet and natural boundary condition. We present computational studies of the given problem.  ...  grids to qualitativelycatch the interface evolution of the two-phase flow.  ... 
doi:10.1007/978-3-642-11795-4_6 fatcat:nx4nvuxaxfbcdjknopny53ck5e

Applications of Distributed-Order Fractional Operators: A Review

Wei Ding, Sansit Patnaik, Sai Sidhardh, Fabio Semperlotti
2021 Entropy  
Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems  ...  The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to  ...  The detailed mathematical analysis of a DO advection-diffusion equation with a discrete distribution of orders was presented in [77] .  ... 
doi:10.3390/e23010110 pmid:33467618 pmcid:PMC7830465 fatcat:twhf4c73dbbnhkveflg43r3rz4

Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients [article]

Pin Lyu, Seakweng Vong
2021 arXiv   pre-print
The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time-space dependent variable coefficients is challenging, especially when the classical  ...  In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations  ...  In this work, we consider numerical analysis of the two-dimensional time fractional evolution equations with general time-space dependent variable coefficients: D α t u = Au + f (x, t), x ∈ Ω, t ∈ (0,  ... 
arXiv:2102.09396v2 fatcat:4ee7ormbcfaurauiqzfc54c7j4

The mathematics of finite elements and applications

1989 International Journal of Engineering Science  
We have selected several benchmark problems; with and without geometrical singularities, with and without change of type of the vorticity equation.  ...  of upwinding for the constitutive equations.  ...  We reduce the analysis of convergence of the fully discretized Galerkin method for the hypersingular integral equation Duv to the analysis of the weakly singular integral operator -1.  ... 
doi:10.1016/0020-7225(89)90012-8 fatcat:z6hknn325re4tbssbqqzmcketa

A multiscale approach to hybrid RANS/LES wall modeling within a high‐order discontinuous Galerkin scheme using function enrichment

Benjamin Krank, Martin Kronbichler, Wolfgang A. Wall
2019 International Journal for Numerical Methods in Fluids  
This is done by providing the Galerkin method with an independent set of shape functions for each of these two methods; the standard high-order polynomial basis resolves turbulent eddies where the mesh  ...  Numerical tests show the outstanding characteristics of the wall model regarding grid independence, superiority compared to equilibrium wall models in separated flows, and achieve a speed-up by two orders  ...  The present section gives an overview of the temporal discretization (Section 4.1) and the spatial operators of the variational form (Section 4.2), and presents a numerical stability analysis of the viscous  ... 
doi:10.1002/fld.4712 fatcat:qe5vfy4szvfltekx76awf2xm2y

A thermal-fully hydrodynamic model for semiconductor devices and applications to III-V HBT simulation

A. Benvenuti, W.M. Coughrau, M.R. Pinto
1997 IEEE Transactions on Electron Devices  
We apply a one-dimensional (1-D) implementation of such a model to the simulation of AlGaAs/GaAs and InP/InGaAs Heterojunction Bipolar Transistors (HBT's), comparing the results with those provided by  ...  Because of the interaction between self-heating and hot carriers effects, neither isothermal nor conventional macrothermal models are adequate for the simulation of state-of-the-art power devices; instead  ...  Naldi of Politecnico di Torino, for their comments and suggestions on the thermal analysis, and to A. Grinberg of Lucent Technologies and S.  ... 
doi:10.1109/16.622585 fatcat:6z4uwpgmy5cnjbxodwtktgfvym

Efficient reordered nonlinear Gauss–Seidel solvers with higher order for black-oil models

Øystein S. Klemetsdal, Atgeirr F. Rasmussen, Olav Møyner, Knut-Andreas Lie
2019 Computational Geosciences  
We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of the open-source OPM Flow simulator.  ...  solver for the transport problems.  ...  Møyner is funded by VISTA, a basic research program funded by Equinor and conducted in close collaboration with the Norwegian Academy of Science and Letters.  ... 
doi:10.1007/s10596-019-09844-5 fatcat:wuppppvsb5eddfy5a43ajhrwg4

Exponential Convergence of hp-Time-Stepping in Space-Time Discretizations of Parabolic PDEs [article]

Ilaria Perugia, Christoph Schwab, Marco Zank
2022 arXiv   pre-print
Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility.  ...  We combine this semi-discretization in time with first-order, so-called "h-version" Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time  ...  Contributions of the present paper are a weighted analytic, temporal regularity analysis based on the analytic semigroup theory for linear, parabolic evolution equations, for source terms and coefficients  ... 
arXiv:2203.11879v1 fatcat:utbpqeyqq5f7thcdzh7jq3xqsm

Mini-Workshop: Interface Problems in Computational Fluid Dynamics

Noel Walkington, Lutz Tobiska, Eberhard Bänsch
2005 Oberwolfach Reports  
One difficulty common to all of these problems is that the associated interfaces are Lagrangian in character, while the fluid equations are naturally posed in the Eulerian frame.  ...  In this report, we concentrate on an issue of temporally-refined graded space-time meshes.  ...  Two different grids are used for the space discretization. The three advection problems are solved on a fixed, structured grid made out of small cubic cells, using a forward Characteristics method.  ... 
doi:10.4171/owr/2005/08 fatcat:j5ul2znajzec7hb45qqayv7tye
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