Crank-Nicolson SPDMFE Method Based on Two Transformations for Sobolev Equation With Convection Term

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1 43«6 Å Vol43, No6 4 ADVANCES IN MATHEMATICSCHINA Nov, 4 doi: 845/sxjz3b Crank-Nicolson SPDMFE Metod Based on Two Transformations for Sobolev Equation Wit Convection Term DU Yanwei, LIU Yang, LI Hong, TONG Mingwang Scool of Matematical Sciences, Inner Mongolia University, Hoot, Inner Mongolia,, P R Cina Abstract: In tis article, we propose and discuss a new splitting positive definite mixed finite element SPDMFE metod for second-order Sobolev equation wit convection term We introduce two transformations: q = u t and σ = ax u + bx u t and solve te ordinary differential equation σ = ax u+bx u t for u, ten reduce te Sobolev equation to a firstorder integro-differential system wit tree variables In te integro-differential system, te equation for te actual stress σ is independent, symmetric, positive definite, and can be solved independentlyfrom bottevariableuandq = u t, tenwe canapproximate tescalar unknown u and te variable q We derive a priori error estimates and stability for bot semidiscrete and Crank-Nicolson fully discrete scemes Finally, we provide some numerical results to illustrate te efficiency of new SPDMFE metod Keywords: Sobolev equation; SPDMFE metod; Crank-Nicolson sceme; transformation; error estimate MR Subject Classification: 65M6; 65N3 / CLC number: O48 Document code: A Article ID: Introduction Yang [3] in proposed a splitting positive definite mixed finite element SPDMFE procedure to treat te pressure equation of parabolic type in a nonlinear parabolic system describing a model for compressible flow displacement in a porous medium Compared to standard mixed metods wose numerical solutions are quite difficult to obtain because of losing positive definite properties, te proposed one does not lead to some saddle point problems From ten on, te metod as been applied to solving some partial differential equations, suc as yperbolic equations [4], pseudo-yperbolic equations [7], viscoelasticity wave equation [5] and integro-differential equations [9,,6] In tis article, we propose and analyze a new splitting mixed finite element sceme based on two transformationsq = u t and z = ax u+bx u t, see [7] and Yang s SPDMFE sceme [3] Received date: 3-- Revised date: Foundation item: Supported by NSFC No 358, No 3635, No 6, Natural Science Fund of Inner Mongolia Autonomous Region No MS8, No MS6, Scientific Researc Projection of Higer Scools of Inner Mongolia No NJZZ, No NJZY399 and Program of Higer-level Talents of Inner Mongolia University No 59 matliuyang@aliyuncom; smsl@imueducn

2 87 43«Te proposed procedure can be split into tree independent symmetric positive definite integrodifferential sub-scemes and does not need to solve a coupled system of equations To sow te teoretical analysis for our metod, we consider te following initial-boundary value problem for Sobolev equation wit convection term u t +cx u ax u+bx u t = fx,t, x,t Ω J, ux,t =, x,t Ω J, ux, = u x, x Ω, were Ω is a bounded convex polygonal domain in R d d 3 wit a smoot boundary Ω, J =,T] is te time interval wit < T < And ux,t represents te displacement, u t = u t u x and fx,t are given functions, coefficients a = ax, b = bx are smoot and bounded functions, coefficient cx = c x,c x,,c d x is a bounded vector, and H : < a ax a < + ; H : < b bx b < + ; d H 3 : < c i x c+ i= for some positive constants a, a, b, b and c Sobolev equations ave many applications in a lot of pysical problems, suc as transport problems of umidity in soil, eat conduction problems in different mediums and porous teories concerned wit percolation into rocks wit cracks Some numerical metods for Sobolev equations, suc as finite element metods [4], mixed finite element metods [6,,3, ], leastsquaresmetods [8,,5] and discontinuousgalerkin metods [5,,6], werestudied and analyzed Te layout of te article is as follows In Section, te new mixed weak formulation and semi-discrete sceme are formulated Error estimates are derived for semi-discrete problems in Section Fully discrete error estimates based on Crank-Nicolson sceme are derived in Section 3 In Section 4, some numerical results are proposed to illustrate te efficiency of te new SPDMFE metod Finally, we give some concluding remarks about te new SPDMFE metod in Section 5 Trougout tis article, C denotes a generic positive constant wic does not depend on te spatial mes parameters u, σ and time-discretization parameter δ and may be different at teir occurrences Usual definitions, notations, and norms of Sobolev spaces as in Refs [4, 8] are used We denote te natural inner product in L Ω or [L Ω] d by, wit norm L Ω or L Ω At te same time, we denote te function space W = Hdiv;Ω = {ω [L Ω] d ; ω L Ω} Splitting Positive Definite Mixed Sceme To formulate te SPDMFE sceme, we first introduce two auxiliary variables [7] q = u t te velocity and σ = ax u+bx u t te actual stress

3 6, Æ,, : Crank-Nicolson SPDMFE Metod for Sobolev Equation 87 Solve te ordinary differential equation σ = ax u+bx u t for u to obtain u = b e at b Differentiating wit respect to time t and using q = u t, we obtain q = u t = b σ a b e at b e aτ b σdτ + u e at b e aτ b σdτ a b u e at b Tus, we obtain te equivalent coupled integral-differential system for te problem a q σ +c γ t β γsσds+ u = fx,t, x,t Ω J, b q +βσ = αγ t β γsσds+ u, x,t Ω J, c u t q =, x,t Ω J, were α = ax bx, β = bx and γt = γx,t = eαt Ten, te following mixed weak formulation of 3 can be given by a q,v σ,v+ c γ t β γsσds+ u,v = fx,t,v, v L Ω, b q, ω+βσ,ω = c u t,v q,v =, αγ t β γsσds+ u,ω, ω W, 3 v L Ω 4 Taking v = ω in 4a for ω W and ten substituting it into 4b, we derive an equivalent mixed weak formulation of te system 4: a βσ,ω+ σ, ω c γ tβ γsσds, ω αβγ t γsσds, ω = fx,t, ω+c γ t u, ω+αγ t u,ω, ω W, b q,v σ,v+ c βγ t γsσds, v = fx,t,v c γ t u,v, v L Ω, c u t,v q,v =, v L Ω 5 Error Estimates for Semi-discrete Sceme Te Stability for Semi-discrete Sceme Let T u and T σ be two families of quasi-regular partitions of te domain Ω, wic may be te same one or not, suc tat te elements in te partitions ave te diameters bounded by u and σ, respectively

4 87 43«Let X u L Ω and V σ W be finite element spaces defined on te partitions T u and T σ, wit te inverse property [4] and te following approximation properties [4,9] : for p + and r,r,k positive integers, inf ω ω L ω V p Ω C r+ σ ω W r+,p Ω, ω Hdiv;Ω [W r+,p Ω] d, σ inf ω ω Lp Ω C r σ ω W Ω, ω Hdiv;Ω [W r+,p Ω] d, ω V r,p σ inf v v Lp Ω C k+ u v W v X k+,p Ω, v L Ω W k+,p Ω, u were r = r+ for te Brezzi-Douglas-Fortin-Marinispaces [,3], r = r for te Brezzi-Douglas- Marini spaces [ 3] Now te semidiscrete SPDMFE metod for 5 consists in determining u,q,σ X u X u V σ suc tat a βσ,ω + σ, ω c γ tβ γsσ ds, ω αβγ t γsσ ds,ω = fx,t, ω +c γ t u, ω +αγ t u,ω, ω V σ, b q,v σ,v + c βγ t γsσ ds,v = fx,t,v c γ t u,v, v X u, c u t,v q,v =, v X u Teorem Te following results of stability for sceme old: a u L Ω C u L Ω + u L Ω + f L Ω + f L Ω ds, b q L Ω + σ L Ω + σ L Ω C u L Ω + f L Ω + f L Ω ds Proof Coose ω = σ in a and apply Caucy-Scwarz inequality and Young inequality to obtain β σ L Ω + σ L Ω t αβγ t γsσ ds + αγ t u L Ω σ L Ω L Ω t + c γ tβ γsσ ds + c γ t u L Ω + f L Ω σ L Ω L Ω C σ L Ω ds+c u L Ω + f L Ω Using Gronwall lemma for 3, we obtain σ L Ω + σ L Ω C u L Ω + f L Ω + + β σ L Ω + σ L Ω 3 f L Ω ds 4 Coosing v = q in b and applying 4, we obtain q L Ω C u L Ω + f L Ω + f L Ω ds 5

5 6, Æ,, : Crank-Nicolson SPDMFE Metod for Sobolev Equation 873 Coosing v = u in c and using 5, we ave d t dt u L Ω C q L Ω u C LΩ + f LΩ + f L Ω ds 6 Integrating 6 wit respect to time from to t, we obtain u L Ω C u L Ω + u L Ω + f L Ω + f L Ω ds 7 Combine 4, 5 and 7 wit triangle inequality to complete te proof Remark It is easy to see tat Eq a is only associated wit te approximate value σ of te actual stress σ So σ, wic does not depend on u and q, can be obtained by solving Eq a Ten q te approximate value of te velocity q and u can be obtained by solving Eq b and c Remark By a similar metod to te one in [6], te existence and uniqueness of te solution for te system can be proved A Priori Error Estimates For our subsequent error analysis, we introduce two operators It is well known tat, in any one of te classical mixed finite element spaces, tere exists an operator R from Hdiv;Ω onto V σ, see Refs [, 9], suc tat, for p +, σ R σ,φ =, φ V σ = {φ = ω,ω V σ }; σ R σ L p Ω C r+ σ σ W r+,p Ω; σ R σ L p Ω Cσ r σ W r,p Ω We also define te L -project operator P from L Ω onto X u suc tat 8 v P v,v =, v L Ω, v X u ; v P v L Ω C k+ u v H k+ Ω, v H k+ Ω 9 Using te definitions of te operators R and P, we can easily obtain te following lemma Lemma Assume tat te solution of system 5 as regular properties tat u t H k+ Ω Ten we ave te following estimates u P u t L Ω C k+ u u t H k+ Ω Let u u = u P u+p u u = λ+θ, q q = q P q+p q q = η +ς, σ σ = σ R σ+r σ σ = ρ+ξ

6 874 43«Subtracting from 5 and using projections 8 and 9, we obtain a βξ,ω + ξ, ω c γ tβ γsξds, ω αβγ t γsξds,ω = βρ,ω + c γ tβ γsρds, ω + αβγ t γsρds,ω, ω V σ, b ς,v σ σ,v + c βγ t γsσ σ ds,v =, v X u, c θ t,v ς,v = λ t,v, v X u, Teorem Assume tat u = P u and te solution of te system 5 as regular properties tat σ L H r+ Ω and u,q,u t L H k+ Ω Ten we ave te error estimates a σ σ L L Ω C r+ σ, b σ σ L L Ω C r σ, c u u L L Ω + q q L L Ω C r σ +k+ u Proof Coose ω = ξ in a and apply Caucy-Scwarz inequality and Young inequality to obtain β ξ L Ω + ξ L Ω t αβγ t γsρds + αβγ t γsξds + βρ L Ω ξ L Ω L Ω L Ω t + c γ tβ γsξds + c γ tβ γsρds ξ L Ω L Ω { [ α β t e αt e αt α +β ρ L Ω } ξ L Ω + α [ β c e αt e αt α ρ L Ω ds + ρ L Ω ds + Cα,α,β,β, ce α αt e αt ρ L Ω ds+ + β β ρ L Ω + β ξ L Ω + ξ L Ω, were < α α α <, < β β β < From, we ave L Ω ξ L Ω ds ] ] ξ L Ω ds ξ L Ω ξ L Ω ds β ξ L Ω + ξ L Ω Cα,α,β,β, ce α αt e αt ρ L Ω ds+ ξ L Ω ds +β β ρ L Ω

7 6, Æ,, : Crank-Nicolson SPDMFE Metod for Sobolev Equation 875 Using Gronwall lemma for, we obtain β ξ L Ω + ξ L Ω Cα,α,β,β, ce α αt e αt ρ L Ω ds+ ρ L Ω Coosing v = ς in b and applying te Caucy-Scwarz inequality, we obtain ς L Ω σ σ L Ω + c βγ t γsσ σ ds ξ L Ω + c βγ t γsξds L Ω + ρ L Ω + c βγ t γsρds L Ω 3 4 Substitute 3 into 4 to obtain ς L Ω ρ L Ω + Cα,α,β,β, ce α αt e αt ρ 5 L Ω ds+ ρ L Ω Coosing v = θ in c and using Caucy-Scwarz inequality, we ave θ L Ω d dt θ L Ω = d dt θ L Ω ς L Ω + λ t L Ω θ L Ω 6 From 6, we ave d dt θ L Ω ς L Ω + λ t L Ω 7 Integrating 7 wit respect to time from to t, we obtain θ L Ω θ L Ω + ς L Ω + λ t L Ωds 8 Substituting 5 into 7 and noting te fact tat θ = P u u =, we get θ L Ω ρ L Ω + λ t L Ωds + Cα,α,β,β, ce α αt e αt ρ L Ω ds 9 Combining 3, 5, 9 and Lemma, we use te triangle inequality to complete te proof 3 Fully Discrete Error Estimates Based on Crank-Nicolson Sceme In tis section, we get te error estimates of fully discrete scemes For te Crank-Nicolson procedure, let = t < t < t < < t M = T be a given partition of te time interval [,T]

8 876 43«wit step lengt t n = nδ, δ = T M for some positive integer M For a smoot function φ on [,T], define φ = φtn+φt and t φ n = φn φ δ For approximating te integrals, we use te composite left rectangle rule n n I n φ δ φ j φsds Eq 5 as te following equivalent formulation a βσ,ω+ σ, ω c βγ t I γσ+c βγ t n I n γσ, ω αβγ t I γσ+αβγ t n I n γσ,ω = f, ω+ c γ t n+γ t u, ω + α γ t n+γ t u,ω b q,v σ,v c βγ t I γσ+c βγ t n I n γσ,v = f,v+ c γ t n+γ t u,v +R 3,v, v L Ω, c t u n,v q,v = R 4,v, v L Ω, 3 were R +R,ω+R, ω, ω W, = β σ σt αβγ t I γσ+αβγ t n I n γσ R αβγ t γσds+αβγ t n n γσds α γ t n+γ t u αγ t u R +R +R 3, = σ σt c βγ t I γσ+c βγ t n I n γσ c βγ t γσds+c βγ t n n γσds c γ t n+γ t u c γ t u R +R +R 3, 3 33

9 6, Æ,, : Crank-Nicolson SPDMFE Metod for Sobolev Equation 877 R 3 = q qt σ σt c βγ t I γσ+c βγ t n I n γσ c βγ t γσds+c βγ t n n γσds c γ t n+γ t u c γ t u R 3 +R, 34 R 4 = t u n u t t q qt 35 R 4 R 3 Now we can formulate a fully discrete sceme: Find u n,qn,σn X u X u V σ n =,,,M suc tat a βσ,ω + σ, ω c βγ t I γσ +c βγ t n I n γσ, ω αβγ t I γσ +αβγ t n I n γσ,ω = f, ω + c γ t n+γ t u, ω + α γ t n+γ t u,ω, ω V σ, 36 b q,v σ,v c βγ t I γσ +c βγ t n I n γσ,v = f,v + c γ t n+γ t u,v, v X u, c t u n,v q,v =, v X u Teorem 3 Te Crank-Nicolson fully discrete sceme 36 satisfies te following inequalities of stability a q L Ω + σ L Ω + σ L Ω n C δ f j + f L + u Ω L Ω L Ω, b u n L Ω C n δ Proof Set w = σ f j + f L + u Ω L Ω L Ω + u L Ω in 36a and apply Caucy-Scwarz inequality and Young in-

10 878 43«equality to obtain β σ L Ω + σ L Ω αβγ t I γσ +αβγ t n I n γσ L Ω + αγ t n+γ t L u σ L Ω Ω + c γ t n+γ t u + L f L Ω Ω + c βγ t I γσ +c βγ t n I n γσ n C δ j σ L Ω + f L Ω + u L Ω + β σ L Ω + σ We use discrete Gronwall lemma to get L Ω L Ω σ σ L Ω + σ L Ω n C δ f j L Ω + f L Ω + u L Ω L Ω Taking v = q in 36b and applying Caucy-Scwarz inequality, Young inequality and 38, we ave q L Ω C n δ f j L Ω + f L Ω + u L Ω 39 Setting v = u 39, we obtain in 36c and applying Caucy-Scwarz inequality, Young inequality and u n L Ω u δ L Ω q C L Ω + u L Ω n C δ f j L Ω + f + u L Ω + u L Ω L Ω 3 Multiplying 3 by δ, summing from to n and using discrete Gronwall lemma, we obtain u n L Ω C n δ f j L Ω + f L Ω + u L Ω + u 3 L Ω

11 6, Æ,, : Crank-Nicolson SPDMFE Metod for Sobolev Equation 879 Remark 3 Usingasimilarmetodasin[],teexistenceanduniquenessoftesolution for te fully discrete system 36 can be proved For fully discrete error estimates, we now split te errors as u n u n = un P u n +P u n u n = λn +θ n, q n q n = q n P q n +P q n q n = η n +ς n, σ n σ n = σn R σ n +R σ n σ n = ρn +ξ n From 3 36, we ten obtain a βξ c βγ t I γξ+c βγ t n I n γξ,ω, ω αβγ t I γξ+αβγ t + ξ n I n γξ, ω,ω c βγ t I γρ+c βγ t n I n γρ =, ω αβγ t I γρ+αβγ t n I n γρ +,ω βρ,ω +R,ω +R, ω, ω V σ, b ς,v σ σ,v c βγ t I γσ I γσ,v c βγ tn I n γσ I n γσ,v 3 = η,v +R 3,v, v X u, c t θ n,v ς,v = t λ n,v +η,v +R 4,v, v X u Lemma 3 Assume tat te solution of te system 5 as regular properties tat u t H k+ Ω Ten we ave te estimates max tu P u n L n M Ω C k+ u Teorem 3 Assume tat 3 u t 3, q t L L Ω, σ t, σ t L L Ω and u = P u Ten tere exists a constant C suc tat a max n M σt σ L Ω C r+ σ +δ, b max n M σt σ L Ω C r σ +δ, c max n M ut n u n L Ω C k+ u + r σ +δ, d max n M qt q L Ω C k+ u + r σ +δ Proof Set w = ξ in 3a and apply Caucy-Scwarz inequality and Young in-

12 88 43«equality to obtain β ξ L Ω + ξ L Ω βρ + αβγ t I γξ+αβγ t n I n γξ L Ω L Ω + αβγ t I γρ+αβγ t n I n γρ + R ξ L Ω L Ω c βγ t I γξ+c βγ t n I n γξ + + R L Ω L Ω + c βγ t I γρ+c βγ t n I n γρ ξ L Ω L Ω [ n ξ Cα,α,β,β, c δ j L Ω + ρ j + L Ω R L Ω + ] R + β L Ω ξ L Ω + ξ L Ω From 33, we obtain β ξ Cα,α,β,β, cδ L Ω + ξ L Ω Cα,α,β,β, c δ ξ j n L Ω +δ ρ j + R L Ω + R L Ω L Ω L Ω Using discrete Gronwall lemma, we ave β Cα,α,β,β, cδ ξ + ξ L Ω L Ω n Cα,α,β,β, c δ ρ j L Ω + R L Ω + R L Ω 35 Taking v = ς in 3b and applying Caucy-Scwarz inequality, Young inequality and 35, we ave ς L Ω σ C σ L Ω + η L Ω + c βγ t I γσ I γσ L Ω + c βγ t n I n γσ I n γσ L Ω + R 3 n C δ ρ j L Ω + R L Ω + R + R 3 L Ω + η L Ω + ρ L Ω L Ω L Ω 36

13 6, Æ,, : Crank-Nicolson SPDMFE Metod for Sobolev Equation 88 Taking v = θ in 3c and applying Caucy-Scwarz inequality and Young inequality, we ave Substitute 36 into 37 to obtain θ n L Ω θ L Ω = t θ n,θ δ ς L Ω + tλ n L Ω + η L Ω + R 4 L Ω + θ L Ω 37 θ n L Ω θ L Ω δ n C δ ρ j + R 3 L Ω + R + R L Ω L Ω + η L Ω + ρ L Ω + θ + t λ n L Ω + R 4 L Ω L Ω L Ω 38 Multiplying 38 by δ and summing from to n, we ave Cδ θ n L Ω J θ L Ω +Cδ n δ ρ j L Ω + R n= L Ω + R L Ω + R 3 L Ω + η L Ω + ρ + t λ n L Ω + R J 4 +Cδ θ n L Ω L Ω n= L Ω Taking δ in suc a way tat for < δ δ, Cδ >, we use Gronwall lemma to obtain J θ J L Ω Cδ n δ ρ j L Ω + ρ n= + t λ n L Ω + η L Ω + R + R L Ω + R 3 L Ω + R 4 L Ω L Ω L Ω 39 3 In order to obtain te error estimate, we first estimate t λ n L Ω, R L Ω, R L Ω, R 3 L Ω and R 4 L Ω Use Taylor expansion to obtain R R L Ω R L Ω + R L Ω + R 3 L Ω Cδ σ tt L L Ω + σ t L L Ω + u L L Ω, L Ω R L Ω + R L Ω + R 3 L Ω Cδ σ tt L L Ω + σ t L L Ω + u L L Ω, 3 3

14 88 43«R 3 R 4 L Ω R 3 L Ω + R L Ω Cδ q tt L L Ω + σ tt L L Ω L Ω R 4 + σ t L L Ω + u L L Ω, L Ω + R 3 L Ω Cδ u ttt L L Ω + q tt L L Ω, n t λ n L Ω = t λ t dt δ n n δ ds λ t L Ω dt t t n L Ω = λ t L δ Ω dt t Substitute 3 35 into 35, 36 and 3, respectively, to obtain ξ L Ω + ξ L Ω [ n C δ ρ j L Ω +δ4 σ tt L L Ω + σ tt L L Ω + σ t L L Ω + u L L Ω ], [ ς L Ω C n δ ρ j L Ω + η L Ω + ρ +δ 4 q tt L L Ω + σ tt L L Ω + σ t L L Ω L Ω 37 and J θ J L Ω Cδ n δ + σ tt L L Ω + u L L Ω ], n= ρ j + ρ L + η Ω L Ω L Ω +Cδ 4 q tt L L Ω + σ tt L L Ω + σ t L L Ω + σ tt L L Ω + u tj L L Ω +C λ t L Ω dt Using and Lemma 3, we apply te triangle inequality to complete te proof 4 Numerical Results 38 In tis section, we sow some numerical results to illustrate te efficiency of SPDMFE metod We consider te following Sobolev equation wit initial-boundary value conditions u t + u x 3 u x t u x = π e t sinπx+πe t cosπx, x,t,],], u,t = u,t =, t [,], ux, = sinπx, x [,] 4

15 6, Æ,, : Crank-Nicolson SPDMFE Metod for Sobolev Equation 883 Itiseasytofindtatteexactsolutionforequation4isux,t = e t sinπx Wecoose te piecewise linear functions in spaces and Crank-Nicolson sceme in time and obtain some numerical results by te Matlab procedure In te following analysis, we will use te formulas w w L L = max n t i= wt n,x i w n,i w w for errors and log L L w w for L L convergence rate In Tables 3, we sow a priori error estimates and convergence rates in L L for te scalar unknown u, te velocity q and te actual stress σ, respectively From te numerical results in Tables 3, we can find tat te convergence rate obtained in te numerical experiment is, wic confirms our teoretical analysis, wen te time step and spatial step ratio is 4 tat is = 4 t = t 4 8 u u L L Table L L -errors and convergence rates for u e-4 695e-5 73e-5 435e-6 83e-6 Rate = t 4 8 q q L L Table L L -errors and convergence rates for q e e e-5 89e-5 84e-6 Rate = t 4 8 σ σ L L Table 3 L L -errors and convergence rates for σ e e e e-7 5e-7 Rate In Figures 3, we describe te comparisons of te numerical solution and te exact solution at t = 5,5,75, wit te spatial mes parameter = and time-discretization parameter t = 4 From te above data obtained in te numerical experiment, we find tat our metod is efficient for second-order Sobolev equation 5 Concluding Remarks In tis article, we study and discuss a new SPDMFE metod for Sobolev equation wit convection-term and derive some a priori error estimates based on bot semi-discrete and Crank- Nicolson fully discrete scemes Compared to standard mixed metods, te proposed metod as some attractive features First, te proposed one does not cause some saddle point problems Second, te Ladysenskaja-Babu ska-brezzi LBB consistency condition between te mixed element spaces X u and V σ is not necessary Moreover, te proposed procedure can be split into

16 884 43«Figure Comparison of te numerical solution u wit te exact solution u Figure Comparison of te numerical solution q wit te exact solution q

17 6, Æ,, : Crank-Nicolson SPDMFE Metod for Sobolev Equation 885 Figure 3 Comparison of te numerical solution σ wit te exact solution σ tree independent symmetric positive definite integro-differential sub-scemes and does not need to solve a coupled system of equations We propose te new SPDMFE sceme based on two transformations [7] : q = u t and z = ax u +bx u t, wic differs from te mixed system in Ref [3] We can see clearly tat te metod proposed in tis article can solve te following pseudo-yperbolic equations [7] u tt ax,t u+bx,t u t +u t = fx,t, x,t Ω J, ux,t =, x,t Ω J, 5 ux, = u x, u t x, = u x, x Ω Acknowledgements Te autors tank te anonymous referees and editors for teir elpful suggestions and comments, wic greatly improve te article References [] Brezzi, F, Douglas Jr, J, Fortin, M and Marini, LD, Efficient rectangular mixed finite elements in two and tree space variables, RAIRO Modél Mat Anal Numér, 987, 4: [] Brezzi, F, Douglas Jr, J and Marini, LD, Two families of mixed finite elements for second order elliptic problems, Numer Mat, 985, 47: 7-35 [3] Cen, ZX, Finite Element Metods and Teir Applications, Berlin: Springer-Verlag, 5 [4] Ciarlet, PG, Te Finite Element Metods for Elliptic Problems, New York: Nort-Holland, 978 [5] Gao, FZ, Qiu, JX and Zang, Q, Local discontinuous Galerkin finite element metod and error estimates for one class of Sobolev equation, J Sci Comput, 9, 43: [6] Gao, FZ and Rui, HX, Two splitting least-squares mixed element metods for linear Sobolev equations, Mat Numer Sin, 8, 33: 69-8 in Cinese [7] Gao, LP, Liang, D and Zang, B, Error estimates for mixed finite element approximations of te viscoelasticity wave equation, Mat Metods Appl Sci, 4, 77: [8] Gu, HM and Yang, DP, Least-squares mixed finite element metod for Sobolev equations, Indian J Pure Appl Mat,, 35: 55-57

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