×

A semi-discrete finite element method for a class of time-fractional diffusion equations. (English) Zbl 1339.65156

Summary: As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly used in the related mathematical descriptions. These models usually involve long-time-range computation, which is a critical obstacle for their application; improvement of computational efficiency is of great significance. In this paper, a semi-discrete method is presented for solving a class of time-fractional diffusion equations that overcome the critical long-time-range computation problem. In the procedure, the spatial domain is discretized by the finite element method, which reduces the fractional diffusion equations to approximate fractional relaxation equations. As analytical solutions exist for the latter equations, the burden arising from long-time-range computation can effectively be minimized. To illustrate its efficiency and simplicity, four examples are presented. In addition, the method is used to solve the time-fractional advection-diffusion equation characterizing the bromide transport process in a fractured granite aquifer. The prediction closely agrees with the experimental data, and the heavy-tail decay of the anomalous transport process is well represented.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R11 Fractional partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Dagan, Annual Review of Fluid Mechanics 19 (1) pp 183– (1987) · Zbl 0687.76091 · doi:10.1146/annurev.fl.19.010187.001151
[2] TRANSPORT POROUS MED 42 pp 155– (2001) · doi:10.1023/A:1006772716244
[3] PHYS REP 339 pp 1– (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[4] ADV WATER RESOUR 32 pp 561– (2009) · doi:10.1016/j.advwatres.2009.01.008
[5] WATER RESOUR RES 40 pp 12416– (2004)
[6] Water Resources Research 31 (6) pp 1461– (1995) · doi:10.1029/95WR00483
[7] ADV WATER RESOUR 31 pp 1578– (2008) · doi:10.1016/j.advwatres.2008.07.002
[8] PHYS REV E 63 pp 021112– (2001) · doi:10.1103/PhysRevE.63.021112
[9] 388 pp 4586– (2009) · doi:10.1016/j.physa.2009.07.024
[10] ADV WATER RESOUR 30 pp 1205– (2007) · doi:10.1016/j.advwatres.2006.11.002
[11] WATER RESOUR RES 36 pp 1403– (2000) · doi:10.1029/2000WR900031
[12] REV GEOPHYS 44 pp 2003– (2006)
[13] ADV WATER RESOUR 30 pp 1408– (2007) · doi:10.1016/j.advwatres.2006.05.029
[14] Magin, Journal of magnetic resonance (San Diego, Calif. : 1997) 190 (2) pp 255– (2008) · doi:10.1016/j.jmr.2007.11.007
[15] CONCEPTS MAGN RESON A 34 pp 16– (2009)
[16] Chaos (Woodbury, N.Y.) 15 pp 026103– (2005) · Zbl 1080.82022 · doi:10.1063/1.1860472
[17] PHYS REP 371 pp 461– (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[18] 22 pp 1230014– (2012) · Zbl 1258.34018 · doi:10.1142/S0218127412300145
[19] 62 pp 855– (2011) · Zbl 1228.65190 · doi:10.1016/j.camwa.2011.02.045
[20] J COMPUT PHYS 228 pp 3137– (2009) · Zbl 1160.65308 · doi:10.1016/j.jcp.2009.01.014
[21] J COMPUT APPL MATH 193 pp 243– (2006) · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005
[22] 22 pp 1250085– (2012) · Zbl 1258.65079 · doi:10.1142/S021812741250085X
[23] NUMER ALGORITHMS 26 pp 336– (2001)
[24] ADV WATER RESOUR 34 pp 47– (2011) · doi:10.1016/j.advwatres.2010.09.012
[25] J COMPUT PHYS 227 pp 886– (2007) · Zbl 1165.65053 · doi:10.1016/j.jcp.2007.05.012
[26] J COMPUT PHYS 216 pp 264– (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006
[27] 48 pp 1017– (2004) · Zbl 1069.65094 · doi:10.1016/j.camwa.2004.10.003
[28] SIAM J NUMER ANAL 47 pp 204– (2008)
[29] APPL MATH COMPUT 217 pp 2534– (2010) · Zbl 1206.65234 · doi:10.1016/j.amc.2010.07.066
[30] 59 pp 1718– (2010) · Zbl 1189.65288 · doi:10.1016/j.camwa.2009.08.071
[31] J COMPUT PHYS 230 pp 3352– (2011) · Zbl 1218.65070 · doi:10.1016/j.jcp.2011.01.030
[32] 33 pp 1159– (2011) · Zbl 1229.35315 · doi:10.1137/100800634
[33] SIGNAL PROCESS 86 pp 2602– (2006) · Zbl 1172.94436 · doi:10.1016/j.sigpro.2006.02.007
[34] WATER RESOUR RES 6 pp 204– (1970) · doi:10.1029/WR006i001p00204
[35] 7 pp 1461– (1996) · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5
[36] PHYS LETT A 373 pp 4405– (2009) · Zbl 1234.65034 · doi:10.1016/j.physleta.2009.10.004
[37] Ground Water 37 (5) pp 770– (1999) · doi:10.1111/j.1745-6584.1999.tb01170.x
[38] WATER RESOUR RES 39 pp 1356– (2003)
[39] J COMPUT APPL MATH 172 pp 65– (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[40] COMMUN NUMER METH ENG 24 pp 2047– (2008) · Zbl 1152.76034 · doi:10.1002/cnm.1094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.