Skip to main content
OSTI.GOVU.S. Department of Energy Office of Scientific and Technical Information
U.S. Department of Energy
Office of Scientific and Technical Information
 

Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes

Journal Article · · Mathematics in Engineering
DOI:https://doi.org/10.3934/mine.2025005· OSTI ID:2570773
 [1];  [2];  [1];  [2];  [3]
  1. Vellore Institute of Technology (India)
  2. Indian Inst. of Technology (IIT), Madras (India)
  3. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
+ Show Author Affiliations

We extend the Virtual Element Method to a two-dimensional unsteady nonlinear convection-diffusion equation characterized by a fractional-order derivative with respect to the time variable. Our methodology is based on three fundamental technical components: a fractional version of the Grunwald-Letnikov approximation, discrete maximal regularity, and the regularity theory associated with non-linearity. We prove the method's well-posedness, i.e., the approximate solution's existence and uniqueness to the time-fractional convection-diffusion equation with a Lipschitz nonlinear source term. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L2-norm and H1-norm to various mesh configurations is validated by numerical results, underlining the practical effectiveness of the proposed method.

Research Organization:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program
Grant/Contract Number:
89233218CNA000001
OSTI ID:
2570773
Report Number(s):
LA-UR--24-22237; 10.3934/mine.2025005
Journal Information:
Mathematics in Engineering, Journal Name: Mathematics in Engineering Journal Issue: 2 Vol. 7; ISSN 2640-3501
Publisher:
AIMS PressCopyright Statement
Country of Publication:
United States
Language:
English

References (45)

Hourglass stabilization and the virtual element method: Hourglass stabilization and the virtual element method journal January 2015
Variational formulation for the stationary fractional advection dispersion equation journal January 2006
Post processing of solution and flux for the nodal mimetic finite difference method: Post Processing of Nodal MFD Method
  • Beirão da Veiga, Lourenço; Manzini, Gianmarco; Putti, Mario
  • Numerical Methods for Partial Differential Equations, Vol. 31, Issue 1 https://doi.org/10.1002/num.21907
journal July 2014
Ill-conditioning in the virtual element method: Stabilizations and bases journal March 2018
Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation
  • Kumar, Dileep; Chaudhary, Sudhakar; Srinivas Kumar, V. V. K.
  • Numerical Methods for Partial Differential Equations, Vol. 35, Issue 6 https://doi.org/10.1002/num.22399
journal June 2019
Some error analysis on virtual element methods journal February 2018
A local projection stabilization virtual element method for the time-fractional Burgers equation with high Reynolds numbers journal January 2023
Virtual element method for semilinear elliptic problems on polygonal meshes journal November 2019
Conforming virtual element approximations of the two-dimensional Stokes problem journal November 2022
The Grünwald–Letnikov method for fractional differential equations journal August 2011
Equivalent projectors for virtual element methods journal September 2013
A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian journal August 2017
The conforming virtual element method for polyharmonic problems journal April 2020
The virtual element method for the time fractional convection diffusion reaction equation with non-smooth data journal March 2022
Finite difference approximations for a fractional advection diffusion problem journal June 2009
Mimetic finite difference method journal January 2014
A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation journal September 2014
Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique journal June 2020
Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking journal January 2014
Mixed virtual element methods for general second order elliptic problems on polygonal meshes journal May 2016
The nonconforming virtual element method journal May 2016
Divergence free virtual elements for the stokes problem on polygonal meshes journal February 2017
Mimetic finite differences for elliptic problems journal December 2008
Relaxation in filled polymers: A fractional calculus approach journal October 1995
Virtual element method for nonlinear convection–diffusion–reaction equation on polygonal meshes journal November 2020
Higher order Grünwald approximations of fractional derivatives and fractional powers of operators journal September 2014
Conforming and nonconforming virtual element methods for elliptic problems journal August 2016
Linear Models of Dissipation whose Q is almost Frequency Independent--II journal November 1967
Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes journal January 2011
A Petrov--Galerkin Finite Element Method for Fractional Convection-Diffusion Equations journal January 2016
A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations journal January 2017
The NonConforming Virtual Element Method for the Stokes Equations journal January 2016
Numerical Analysis of Nonlinear Subdiffusion Equations journal January 2018
Convergence Analysis of the mimetic Finite Difference Method for Elliptic Problems with Staggered Discretizations of Diffusion Coefficients journal January 2017
Basic Principles of Virtual Element Methods journal November 2012
New perspectives on polygonal and polyhedral finite element methods journal May 2014
Virtual Element Method for general second-order elliptic problems on polygonal meshes journal February 2016
The nonconforming virtual element method for plate bending problems journal July 2016
Stability analysis for the virtual element method journal October 2017
The fully nonconforming virtual element method for biharmonic problems journal December 2017
Virtual element methods on meshes with small edges or faces journal June 2018
Numerical solutions to integral equations equivalent to differential equations with fractional time journal June 2010
A Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations journal April 2013
A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem journal January 2022
Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems journal June 2018

Figures / Tables (32)

Figure 1(p. 20)figure Figure 1
Figure 2(p. 21)figure Figure 2
Figure 3(p. 21)figure Figure 3
Figure 4(p. 21)figure Figure 4
Figure 5(p. 22)figure Figure 5
Table 1(p. 22)table Table 1
Table 2(p. 22)table Table 2
Table 3(p. 23)table Table 3
Table 4(p. 23)table Table 4
Table 5(p. 23)table Table 5
Table 6(p. 23)table Table 6
Table 7(p. 24)table Table 7
Table 8(p. 24)table Table 8
Table 9(p. 24)table Table 9
Figure 6(p. 25)figure Figure 6
Figure 7(p. 25)figure Figure 7
Figure 8(p. 25)figure Figure 8
Figure 9(p. 26)figure Figure 9
Table 10(p. 26)table Table 10
Table 11(p. 26)table Table 11
Table 12(p. 27)table Table 12
Table 13(p. 27)table Table 13
Table 14(p. 27)table Table 14
Table 15(p. 27)table Table 15
Table 16(p. 28)table Table 16
Table 17(p. 28)table Table 17
Table 18(p. 28)table Table 18
Figure 10(p. 29)figure Figure 10
Table 19(p. 29)table Table 19
Table 20(p. 29)table Table 20
Table 21(p. 30)table Table 21
Table 22(p. 30)table Table 22

Similar Records

The Conforming Virtual Element Method for the convection-diffusion-reaction equation with variable coeffcients.
Technical Report · Thu Oct 02 00:00:00 EDT 2014 · OSTI ID:1159207
Fractional space–time nonlinear reaction–convection–diffusion
Journal Article · Sat Sep 15 00:00:00 EDT 2018 · Computational and Applied Mathematics · OSTI ID:22769240
Numerical Method for Fractional Advection-Diffusion Equation with Heredity
Journal Article · Tue May 15 00:00:00 EDT 2018 · Journal of Mathematical Sciences · OSTI ID:22771275

Related Subjects

97 MATHEMATICS AND COMPUTING
Grunwald-Letnikov approximation
Virtual Element Method
projection operator
regularity theory