March  2017, 22(2): 339-368. doi: 10.3934/dcdsb.2017016

Analysis of a nonlocal-in-time parabolic equation

Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

* Corresponding author: Zhi Zhou (zhizhou0125@gmail.com)

Received  March 2016 Revised  July 2016 Published  December 2016

Fund Project: This research is supported in part by the AFOSR MURI center for Material Failure Prediction through peridynamics and the ARO MURI Grant W911NF-15-1-0562

In this paper, we consider an initial boundary value problem for nonlocal-in-time parabolic equations involving a nonlocal in time derivative. We first show the uniqueness and existence of the weak solution of the nonlocal-in-time parabolic equation, and also the spatial smoothing properties. Moreover, we develop a new framework to study the local limit of the nonlocal model as the horizon parameter δ approaches 0. Exploiting the spatial smoothing properties, we develop a semi-discrete scheme using standard Galerkin finite element method for the spatial discretization, and derive error estimates dependent on data smoothness. Finally, extensive numerical results are presented to illustrate our theoretical findings.

Citation: Qiang Du, Jiang Yang, Zhi Zhou. Analysis of a nonlocal-in-time parabolic equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 339-368. doi: 10.3934/dcdsb.2017016
References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, vol. 140, Academic press, 2003. Google Scholar

[2]

M. AllenL. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Archive for Rational Mechanics and Analysis, 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

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E. Askari, F. Bobaru, R. Lehoucq, M. Parks, S. Silling and O. Weckner, Peridynamics for multiscale materials modeling, in Journal of Physics: Conference Series IOP Publishing, 125 (2008), 012078. Google Scholar

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B. BerkowitzJ. KlafterR. Metzler and H. Scher, Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations, Water Resources Research, 38 (2002), 1-9.   Google Scholar

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J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, IOS Press, Amsterdam, (2001), 439-455. Google Scholar

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O. DefterliM. D'EliaQ. DuM. GunzburgerR. Lehoucq and M. M. Meerschaert, Fractional diffusion on bounded domains, Fractional Calculus and Applied Analysis, 18 (2015), 342-360.  doi: 10.1515/fca-2015-0023.  Google Scholar

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Q. DuM. GunzburgerR. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Mathematical Models and Methods in Applied Sciences, 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.  Google Scholar

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Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696.  doi: 10.1137/110833294.  Google Scholar

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Q. DuY. TaoX. Tian and J. Yang, Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients, Comput. Methods Appl. Mech. Engrg., 310 (2016), 605-627.  doi: 10.1016/j.cma.2016.07.023.  Google Scholar

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L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematics Society, 2010. Google Scholar

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G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

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M. GionaS. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A: Statistical Mechanics and its Applications, 191 (1992), 449-453.   Google Scholar

[13]

B. JinR. LazarovJ. Pasciak and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA Journal of Numerical Analysis, 35 (2014), 561-582.  doi: 10.1093/imanum/dru018.  Google Scholar

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B. JinR. Lazarov and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM Journal on Numerical Analysis, 51 (2013), 445-466.  doi: 10.1137/120873984.  Google Scholar

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B. JinR. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA Journal of Numerical Analysis, 36 (2016), 197-221.  doi: 10.1093/imanum/dru063.  Google Scholar

[16]

B. JinR. Lazarov and Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM Journal on Scientific Computing, 38 (2016), A146-A170.  doi: 10.1137/140979563.  Google Scholar

[17]

Y. L. Keung and J. Zou, Numerical identifications of parameters in parabolic systems, Inverse Problems, 14 (1998), 83-100.  doi: 10.1088/0266-5611/14/1/009.  Google Scholar

[18]

A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, Amsterdam, 2006. Google Scholar

[19]

B. Kilic and E. Madenci, Coupling of peridynamic theory and the finite element method, Journal of Mechanics of Materials and Structures, 5 (2010), 707-733.   Google Scholar

[20]

C. LubichI. Sloan and V. Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Mathematics of Computation of the American Mathematical Society, 65 (1996), 1-17.  doi: 10.1090/S0025-5718-96-00677-1.  Google Scholar

[21]

W. McLean and K. Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293 (2015), 201-217.  doi: 10.1016/j.jcp.2014.08.050.  Google Scholar

[22]

T. Mengesha and Q. Du, Characterization of function spaces of vector fields and an application in nonlinear peridynamics, Nonlinear Analysis A: Theory, Methods and Applications, 140 (2016), 82-111.  doi: 10.1016/j.na.2016.02.024.  Google Scholar

[23]

T. Mengesha and D. Spector, Localization of nonlocal gradients in various topologies, Calculus of Variations and Partial Differential Equations, 52 (2015), 253-279.  doi: 10.1007/s00526-014-0711-3.  Google Scholar

[24]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[25]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[26]

S. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, Journal of the Mechanics and Physics of Solids, 48 (2000), 175-209.  doi: 10.1016/S0022-5096(99)00029-0.  Google Scholar

[27]

S. Silling and R. Lehoucq, Peridynamic theory of solid mechanics, Advances in Applied Mechanics, 44 (2010), 73-166.   Google Scholar

[28]

S. SillingO. WecknerE. Askari and F. Bobaru, Crack nucleation in a peridynamic solid, International Journal of Fracture, 162 (2010), 219-227.   Google Scholar

[29]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, vol. 25 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2006. Google Scholar

[30]

X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM Journal on Numerical Analysis, 52 (2014), 1641-1665.  doi: 10.1137/130942644.  Google Scholar

[31]

J. Xie and J. Zou, Numerical reconstruction of heat fluxes, SIAM Journal on Numerical Analysis, 43 (2005), 1504-1535.  doi: 10.1137/030602551.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, vol. 140, Academic press, 2003. Google Scholar

[2]

M. AllenL. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Archive for Rational Mechanics and Analysis, 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

[3]

E. Askari, F. Bobaru, R. Lehoucq, M. Parks, S. Silling and O. Weckner, Peridynamics for multiscale materials modeling, in Journal of Physics: Conference Series IOP Publishing, 125 (2008), 012078. Google Scholar

[4]

B. BerkowitzJ. KlafterR. Metzler and H. Scher, Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations, Water Resources Research, 38 (2002), 1-9.   Google Scholar

[5]

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, IOS Press, Amsterdam, (2001), 439-455. Google Scholar

[6]

O. DefterliM. D'EliaQ. DuM. GunzburgerR. Lehoucq and M. M. Meerschaert, Fractional diffusion on bounded domains, Fractional Calculus and Applied Analysis, 18 (2015), 342-360.  doi: 10.1515/fca-2015-0023.  Google Scholar

[7]

Q. DuM. GunzburgerR. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Mathematical Models and Methods in Applied Sciences, 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.  Google Scholar

[8]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696.  doi: 10.1137/110833294.  Google Scholar

[9]

Q. DuY. TaoX. Tian and J. Yang, Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients, Comput. Methods Appl. Mech. Engrg., 310 (2016), 605-627.  doi: 10.1016/j.cma.2016.07.023.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematics Society, 2010. Google Scholar

[11]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[12]

M. GionaS. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A: Statistical Mechanics and its Applications, 191 (1992), 449-453.   Google Scholar

[13]

B. JinR. LazarovJ. Pasciak and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA Journal of Numerical Analysis, 35 (2014), 561-582.  doi: 10.1093/imanum/dru018.  Google Scholar

[14]

B. JinR. Lazarov and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM Journal on Numerical Analysis, 51 (2013), 445-466.  doi: 10.1137/120873984.  Google Scholar

[15]

B. JinR. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA Journal of Numerical Analysis, 36 (2016), 197-221.  doi: 10.1093/imanum/dru063.  Google Scholar

[16]

B. JinR. Lazarov and Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM Journal on Scientific Computing, 38 (2016), A146-A170.  doi: 10.1137/140979563.  Google Scholar

[17]

Y. L. Keung and J. Zou, Numerical identifications of parameters in parabolic systems, Inverse Problems, 14 (1998), 83-100.  doi: 10.1088/0266-5611/14/1/009.  Google Scholar

[18]

A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, Amsterdam, 2006. Google Scholar

[19]

B. Kilic and E. Madenci, Coupling of peridynamic theory and the finite element method, Journal of Mechanics of Materials and Structures, 5 (2010), 707-733.   Google Scholar

[20]

C. LubichI. Sloan and V. Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Mathematics of Computation of the American Mathematical Society, 65 (1996), 1-17.  doi: 10.1090/S0025-5718-96-00677-1.  Google Scholar

[21]

W. McLean and K. Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293 (2015), 201-217.  doi: 10.1016/j.jcp.2014.08.050.  Google Scholar

[22]

T. Mengesha and Q. Du, Characterization of function spaces of vector fields and an application in nonlinear peridynamics, Nonlinear Analysis A: Theory, Methods and Applications, 140 (2016), 82-111.  doi: 10.1016/j.na.2016.02.024.  Google Scholar

[23]

T. Mengesha and D. Spector, Localization of nonlocal gradients in various topologies, Calculus of Variations and Partial Differential Equations, 52 (2015), 253-279.  doi: 10.1007/s00526-014-0711-3.  Google Scholar

[24]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[25]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[26]

S. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, Journal of the Mechanics and Physics of Solids, 48 (2000), 175-209.  doi: 10.1016/S0022-5096(99)00029-0.  Google Scholar

[27]

S. Silling and R. Lehoucq, Peridynamic theory of solid mechanics, Advances in Applied Mechanics, 44 (2010), 73-166.   Google Scholar

[28]

S. SillingO. WecknerE. Askari and F. Bobaru, Crack nucleation in a peridynamic solid, International Journal of Fracture, 162 (2010), 219-227.   Google Scholar

[29]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, vol. 25 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2006. Google Scholar

[30]

X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM Journal on Numerical Analysis, 52 (2014), 1641-1665.  doi: 10.1137/130942644.  Google Scholar

[31]

J. Xie and J. Zou, Numerical reconstruction of heat fluxes, SIAM Journal on Numerical Analysis, 43 (2005), 1504-1535.  doi: 10.1137/030602551.  Google Scholar

Figure 1.  Plot of $\|u_h^\delta-u_h^0\|_{L^2(0,T;L^2(\Omega))}$ for example (a), (b) and (c), with $\tau=10^{-4}$, $h=10^{-3}$
Table 1.  $\| u-u_h \|_{L^2(0,T;\dot H^{p}\left( \Omega \right))}$ for example (a) with $\delta=1$, $h=1/N$, $\tau=10^{-5}$, $T=1$
$\gamma$$\text{norm}\backslash N$$10$$20$$40$ $80$ $160$rate
$-0.8$$L^2$4.58e-41.15e-42.87e-57.18e-61.79e-6 $\approx$ 2.00 (2.00)
$H^1$9.42e-34.71e-32.35e-31.18e-35.53e-4 $\approx$ 1.05 (1.00)
$-0.2$$L^2$2.99e-47.51e-51.88e-54.69e-61.17e-6 $\approx$ 2.00 (2.00)
$H^1$5.57e-32.78e-31.39e-36.94e-43.26e-4 $\approx$ 1.05 (1.00)
$0.2$$L^2$2.54e-46.36e-51.59e-53.97e-69.90e-7 $\approx$ 2.00 (2.00)
$H^1$4.59e-32.29e-31.14e-35.72e-42.69e-4 $\approx$ 1.07 (1.00)
$0.8$$L^2$2.22e-45.58e-51.40e-53.49e-68.69e-7$\approx$ 2.00 (2.00)
$H^1$3.98e-31.99e-39.94e-44.97e-42.34e-4 $\approx$ 1.06 (1.00)
$\gamma$$\text{norm}\backslash N$$10$$20$$40$ $80$ $160$rate
$-0.8$$L^2$4.58e-41.15e-42.87e-57.18e-61.79e-6 $\approx$ 2.00 (2.00)
$H^1$9.42e-34.71e-32.35e-31.18e-35.53e-4 $\approx$ 1.05 (1.00)
$-0.2$$L^2$2.99e-47.51e-51.88e-54.69e-61.17e-6 $\approx$ 2.00 (2.00)
$H^1$5.57e-32.78e-31.39e-36.94e-43.26e-4 $\approx$ 1.05 (1.00)
$0.2$$L^2$2.54e-46.36e-51.59e-53.97e-69.90e-7 $\approx$ 2.00 (2.00)
$H^1$4.59e-32.29e-31.14e-35.72e-42.69e-4 $\approx$ 1.07 (1.00)
$0.8$$L^2$2.22e-45.58e-51.40e-53.49e-68.69e-7$\approx$ 2.00 (2.00)
$H^1$3.98e-31.99e-39.94e-44.97e-42.34e-4 $\approx$ 1.06 (1.00)
Table 2.  $\| (u-u_h)(t) \|_{\dot H^{p}\left( \Omega \right)}$ for example (b) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $t=0.5$
$\gamma$$\text{norm}\backslash m$$11$$21$$41$ $81$ $161$ $321$rate
$-0.8$$L^2$2.40e-38.04e-42.72e-49.29e-53.21e-51.10e-5$\approx$ 1.55
$H^1$7.40e-25.10e-23.54e-22.45e-21.66e-21.09e-2$\approx$ 0.55
$-0.2$$L^2$6.04e-32.24e-38.08e-42.88e-41.02e-43.52e-5 $\approx$ 1.48
$H^1$2.22e-11.59e-91.12e-97.86e-25.38e-23.53e-2 $\approx$ 0.53
$0.2$$L^2$7.67e-32.88e-31.05e-33.77e-41.33e-44.63e-5 $\approx$ 1.47
$H^1$2.88e-12.07e-11.47e-11.03e-17.08e-24.66e-2 $\approx$ 0.53
$0.8$$L^2$9.35e-33.56e-31.30e-34.70e-41.66e-45.78e-5 $\approx$ 1.47
$H^1$3.57e-12.58e-11.84e-11.29e-18.50e-25.81e-2$\approx$ 0.52
$\gamma$$\text{norm}\backslash m$$11$$21$$41$ $81$ $161$ $321$rate
$-0.8$$L^2$2.40e-38.04e-42.72e-49.29e-53.21e-51.10e-5$\approx$ 1.55
$H^1$7.40e-25.10e-23.54e-22.45e-21.66e-21.09e-2$\approx$ 0.55
$-0.2$$L^2$6.04e-32.24e-38.08e-42.88e-41.02e-43.52e-5 $\approx$ 1.48
$H^1$2.22e-11.59e-91.12e-97.86e-25.38e-23.53e-2 $\approx$ 0.53
$0.2$$L^2$7.67e-32.88e-31.05e-33.77e-41.33e-44.63e-5 $\approx$ 1.47
$H^1$2.88e-12.07e-11.47e-11.03e-17.08e-24.66e-2 $\approx$ 0.53
$0.8$$L^2$9.35e-33.56e-31.30e-34.70e-41.66e-45.78e-5 $\approx$ 1.47
$H^1$3.57e-12.58e-11.84e-11.29e-18.50e-25.81e-2$\approx$ 0.52
Table 3.  $\| (u-u_h)(t) \|_{\dot H^{p}\left( \Omega \right)}$ for example (b) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $t=1.2$
$\gamma$$\text{norm}\backslash m$$11$$21$$41$ $81$ $161$ $321$rate
$-0.8$$L^2$1.36e-43.75e-59.86e-62.53e-66.37e-71.58e-7 $\approx$ 1.95
$H^1$1.56e-38.09e-44.08e-42.02e-49.77e-54.47e-5 $\approx$ 1.03
$-0.2$$L^2$4.11e-41.13e-42.98e-57.63e-61.93e-64.77e-7 $\approx$ 1.95
$H^1$5.98e-33.10e-31.57e-37.78e-43.76e-41.72e-4 $\approx$ 1.02
$0.2$$L^2$5.22e-41.44e-43.78e-59.68e-62.44e-66.06e-7 $\approx$ 1.95
$H^1$8.01e-34.15e-32.10e-31.04e-45.03e-42.30e-4 $\approx$ 1.02
$0.8$$L^2$6.32e-41.74e-44.58e-51.17e-62.96e-67.34e-7 $\approx$ 1.95
$H^1$1.01e-25.24e-32.65e-31.31e-46.34e-42.90e-4 $\approx$ 1.02
$\gamma$$\text{norm}\backslash m$$11$$21$$41$ $81$ $161$ $321$rate
$-0.8$$L^2$1.36e-43.75e-59.86e-62.53e-66.37e-71.58e-7 $\approx$ 1.95
$H^1$1.56e-38.09e-44.08e-42.02e-49.77e-54.47e-5 $\approx$ 1.03
$-0.2$$L^2$4.11e-41.13e-42.98e-57.63e-61.93e-64.77e-7 $\approx$ 1.95
$H^1$5.98e-33.10e-31.57e-37.78e-43.76e-41.72e-4 $\approx$ 1.02
$0.2$$L^2$5.22e-41.44e-43.78e-59.68e-62.44e-66.06e-7 $\approx$ 1.95
$H^1$8.01e-34.15e-32.10e-31.04e-45.03e-42.30e-4 $\approx$ 1.02
$0.8$$L^2$6.32e-41.74e-44.58e-51.17e-62.96e-67.34e-7 $\approx$ 1.95
$H^1$1.01e-25.24e-32.65e-31.31e-46.34e-42.90e-4 $\approx$ 1.02
Table 4.  $\| (u-u_h)(t) \|_{\dot H^{p}\left( \Omega \right)}$ for example (b) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $t=0.5$
$\gamma$$\text{norm}\backslash m$$10$$20$$40$ $80$ $160$ $320$rate
$-0.8$$L^2$2.88e-47.21e-51.80e-54.51e-61.12e-62.77e-7 $\approx$ 2.00
$H^1$7.88e-33.95e-31.97e-39.87e-44.64e-42.10e-4 $\approx$ 1.05
$0.8$$L^2$6.27e-51.57e-53.94e-69.85e-72.45e-76.07e-8 $\approx$ 2.00
$H^1$1.43e-37.12e-43.56e-41.78e-48.37e-53.78e-5 $\approx$ 1.05
$\gamma$$\text{norm}\backslash m$$10$$20$$40$ $80$ $160$ $320$rate
$-0.8$$L^2$2.88e-47.21e-51.80e-54.51e-61.12e-62.77e-7 $\approx$ 2.00
$H^1$7.88e-33.95e-31.97e-39.87e-44.64e-42.10e-4 $\approx$ 1.05
$0.8$$L^2$6.27e-51.57e-53.94e-69.85e-72.45e-76.07e-8 $\approx$ 2.00
$H^1$1.43e-37.12e-43.56e-41.78e-48.37e-53.78e-5 $\approx$ 1.05
Table 5.  $\| u-u_h \|_{L^2(0,T;\dot H^{p}\left( \Omega \right))}$ for example (c) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $T=1$
$\gamma$$\text{norm}\backslash m$$21$$41$ $81$ $161$ $321$rate
$-0.5$$L^2$4.67e-31.73e-36.23e-42.21e-47.68e-5$\approx$ 1.47 ($1.50$)
$H^1$3.41e-12.43e-11.71e-11.17e-17.71e-2$\approx$ 0.52 ($0.50$)
$0.5$$L^2$4.68e-31.73e-36.23e-42.21e-47.68e-5$\approx$ 1.47 ($1.50$)
$H^1$3.41e-12.43e-11.71e-11.17e-17.72e-2$\approx$ 0.52 ($0.50$)
$\gamma$$\text{norm}\backslash m$$21$$41$ $81$ $161$ $321$rate
$-0.5$$L^2$4.67e-31.73e-36.23e-42.21e-47.68e-5$\approx$ 1.47 ($1.50$)
$H^1$3.41e-12.43e-11.71e-11.17e-17.71e-2$\approx$ 0.52 ($0.50$)
$0.5$$L^2$4.68e-31.73e-36.23e-42.21e-47.68e-5$\approx$ 1.47 ($1.50$)
$H^1$3.41e-12.43e-11.71e-11.17e-17.72e-2$\approx$ 0.52 ($0.50$)
Table 6.  $\| u-u_h \|_{L^2(0,T;\dot H^{p}\left( \Omega \right))}$ for example (c) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $T=1$
$\gamma$$\text{norm}\backslash m$$10$$20$$40$ $80$ $160$ $320$rate
$-0.5$$L^2$2.51e-46.29e-51.57e-53.93e-69.79e-72.42e-7 $\approx$ 2.00
$H^1$9.33e-34.68e-32.34e-31.17e-35.51e-42.49e-4 $\approx$ 1.07
$0.5$$L^2$2.06e-45.18e-51.29e-53.23e-68.06e-71.99e-7 $\approx$ 2.02
$H^1$7.12e-33.57e-31.79e-38.93e-44.20e-41.90e-4 $\approx$ 1.08
$\gamma$$\text{norm}\backslash m$$10$$20$$40$ $80$ $160$ $320$rate
$-0.5$$L^2$2.51e-46.29e-51.57e-53.93e-69.79e-72.42e-7 $\approx$ 2.00
$H^1$9.33e-34.68e-32.34e-31.17e-35.51e-42.49e-4 $\approx$ 1.07
$0.5$$L^2$2.06e-45.18e-51.29e-53.23e-68.06e-71.99e-7 $\approx$ 2.02
$H^1$7.12e-33.57e-31.79e-38.93e-44.20e-41.90e-4 $\approx$ 1.08
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