A High-Order Accurate Wavelet Method for Solving Three-Dimensional Poisson Problems

Xiaojing Liu,Jizeng Wang,Youhe Zhou

A High-Order Accurate Wavelet Method for Solving Three-Dimensional Poisson Problems

Xiaojing Liu1,2,Jizeng Wang1,Youhe Zhou1

Based on the approximation scheme for aL2-function defined on a three-dimensional bounded space by combining techniques of boundary extension and Coiflet-type wavelet expansion,a modified wavelet Galerkin method is proposed for solving three-dimensional Poisson problems with various boundary conditions.Such a wavelet-based solution procedure has been justified by solving five test examples.Numerical results demonstrate that the present wavelet method has an excellent numerical accuracy,a fast convergence rate,and a very good capability in handling complex boundary conditions.

Coiflet-type wavelet;Galerkin method;three-dimensional Poisson equation;mixed boundary conditions.

1 Introduction

The Poisson problem,as a typical elliptic partial differential equation,plays a central role in mathematics,theoretical physics,mechanics and other fields,such as electromagnetics[Heise and Kuhn(1996)],fluid dynamics[Vuik,Segal,and Meijerink(1999)],plasma physics[Feng and Sheng(2015)],and electrical power network modeling[Howle and Vavasis(2005)].It has so broad applications that researchers have to frequently find a numerical solution of the Poisson equation[Doha(1989);Wordelman,Aluru,and Ravaioli(2000);Feng and Sheng(2015)].

In the past few decades,a number of numerical methods have been proposed to solvethe Poissonequations.For example,Mittal and Gahlaut(1987)introduced the second-and fourth-order finite difference schemes for solving Poisson equations.Doha(1989)developed a Chebyshev spectral method to study Poisson problems.A fourth-order compact difference scheme for solving the three-dimensional Poisson equation on a cubic domain has been proposed by Ge(2010).Barad and Colella(2005)used a local refinement finite volume method to solve the Poisson equation.And a Haar wavelet method for solving two-and three-dimensional Poisson problems has been derived by Shi,Cao,and Chen(2012).Although these numerical methods are effectively applied to the solution of Poisson problems,researchers have been trying to find a more high-order accurate method to solve numerically Poisson problems,which allows for using lower computational cost with respect to a low-order accurate method to ensure a similar accuracy[Zhang,(1998);Shi,Cao,and Chen(2012);Feng and Sheng(2015);Bardazzi,Lugni,Antuono,Graziani,and Faltinsen(2015)].

In our recent work[Liu,Zhou,Wang,and Wang(2013);Liu,Wang,and Zhou(2013);Liu,Wang,and Zhou(2013);Liu,Zhou,Zhang,and Wang(2014)],we have developed the modified wavelet Galerkin methods for solutions of one-and two-dimensional nonlinear problems,which have shown an excellent numerical accuracy and a fast convergence rate.For instance,the order of convergence of such wavelet algorithm for two-dimensional Bratu-like equations is about 5[Liu,Zhou,Wang,and Wang(2013)].And for the Burgers’equation,this wavelet method also has a convergence rate of order 5,and shows a much better accuracy than many other existing methods[Liu,Zhou,Zhang,and Wang(2014)],such as the multiquadric method[Hon and Mao(1998)],the multiquadric quasi-interpolation method[Chen and Wu(2006)],the cubic B-spline quasi-interpolation method[Zhu and Wang(2009)],themultiquadric-RBFmethod[Xieand Li(2013)],theweighted average differential quadrature method[Jiwari,Mittal,and Sharma(2013)],the lattice Boltzmann method[Gao,Le,and Shi(2013)],and the B-spline finite element method[Kutluay,Esen,and Dag(2004)].

In the present study,based on the modified wavelet Galerkin methods respectively for the solution of one-and two-dimensional nonlinear boundary value problems[Liu,Zhou,Wang,and Wang(2013);Liu,Wang,and Zhou(2013)],we propose a wavelet approximation scheme for three-dimensional bounded functions based on techniques of boundary extension and Coiflet-type wavelet expansion,which can eliminate the undesired oscillating error near boundary points due to function value jump[Liu,Zhou,Wang,and Wang(2013)].Then a wavelet-based solution procedure for three-dimensional Poisson problems is derived in detail.At last,a comparison between the present solutions and those obtained by using other existing numerical methods is made to demonstrate the effectiveness of the proposed wavelet method.

2 Wavelet approximation of an interval-bounded L2-function

Following the theory of wavelet based multiresolution analysis,a set of scaling basesfor three-dimensional spacecan bedirectly extended by thetensor productsof one-dimensional wavelet bases[Meyer(1992);Ray and Gupta(2014)].Therefore based on our previous work[Liu,Zhou,Wang,and Wang(2013)],for a functionf(x,y,z)∈L2[01]3,we have

in whichjx,jyandjzare the decomposition level respectively in thex,yandzdirections,and the modified one-dimensional wavelet basis

Here φ(x)is the generalized Coiflet-type orthogonal scaling function with first order momentM1=7 and number of vanishing moment β=6 of the corresponding wavelet function,which is developed by Wang[Wang(2001)].And in Eq.(2),expressions

where coefficientsp0,i,kandp1,i,kof numerical differentiation are determined by relationshipsand matrixes[Liu,Zhou,Wang,and Wang(2013);Wang(2014)]

Followingtheerror analysisfor thiswaveletapproximationinone-and two-dimensional problems accomplished by Liu et al.[Liu,Zhou,Zhang,and Wang(2014);Liu,Wang,and Zhou(2013)]and the theory of multiresolution analysis[Meyer(1992)],we can similarly obtain

in which exponent γ=min{jx(β −n),jy(β −m),jz(β −l)},constantsCdepends on the smoothness and boundary extension property off(x,y,z),andn,m,lare non-negative integers satisfyingn,m,l＜β=6.

3 Solution of the three-dimensional poisson equation

In this section,we will propose a modified Galerkin method based on the wavelet approximation(2)to solve the three-dimensional Poisson equation in Cartesian coordinates with the Robin type boundary conditions,as follows:

in whichai,bi,ci,di,i=1,2,3 are constants,andgi,hi,fare known functions.Following the wavelet approximation(1),the unknown functionp(x1,x2,x3)and the source functionf(x1,x2,x3)can be approximated respectively as

Substituting Eqs.(7)and(8)into Eq.(6a),yields

Multiplyingboth sidesof Eq.(9)by ϕj1,k0(x1)ϕj2,l0(x2)ϕj3,n0(x3),k0=1,2,...,2j1−1,l0=1,2,...,2j2−1,n0=1,2,...,2j3−1,respectively and perform integration over the region[0,1]3,gives

On the other hand,substituting Eq.(7)into boundary conditions(6b),we have

Assigningx2=0,1/2j2,...1,x3=0,1/2j3,...1 fori=1,x1=1/2j1,2/2j1,...1−1/2j1,x3=0,1/2j3,...1 fori=2,andx1=1/2j1,2/2j1,...1−1/2j1,x2=1/2j2,2/2j2,...1−1/2j2fori=3,respectively,yields

Similarly,based on boundary conditions(6c)one can obtain

By solving simultaneously Eqs.(10),(12)and(13),we can obtain the nodal values of unknown functionp(k/2j1,l/2j2,n/2j3),k=0,1,...,2j1,l=0,1,...,2j2,n=0,1,...,2j3,which can be used to reconstructp(x1,x2,x3)in terms of Eq.(7).We note that this wavelet solution is valid for the Poisson equation(6a)with almost all the classic types of boundary conditions.For example in Eq.(6b),parametersbi=0 represent the Dirichlet boundary conditions,ai=0 represent the Neumann boundary conditions,and arbitrary values ofaiandbirepresent the general Robin boundary conditions.Moreover by using the similar algorithm,we also can obtain the wavelet solutions of the one-and two-dimensional Poisson problems.

4 Numerical examples

In the following,we will demonstrate the efficiency and accuracy of the proposed wavelet method by numerically solving Poisson equations with various boundary conditions.To effectively evaluate the performance of the present method,we consider the maximum absolute errorL∞,mean absolute errorL1,relative error normL2and order of convergenceR∞,1,which are,respectively,defined as

in whichNandDare the number of grid points and the spatial dimension of the problem,respectively[Barad and Colella(2005);Atluri and Zhu(1998)].

Example1Weconsider theone-dimensional Poisson equationwith Dirichletboundary conditions as follows:

whose exact solution isp(x)= −(1−2x)2sin(2πx)[Gibou,Fedkiw,Cheng,and Kang(2002)].

Table 1:Mean absolute error L1 and order of convergence R1 for problem(18).

Table 2:Maximum absolute error L∞ and order of convergence R∞ for problem(18).

Tables 1 and 2 show the errors of numerical solutions of the one-dimensional Poisson problem(18)given by the proposed wavelet method with various values of grid pointsN.It can be seen from Tables 1 and 2 that results obtained using the present wavelet method with less number of grid pointsNhas a much better numerical accuracy than those given by the finite difference method(FDM)[Gibou,Fedkiw,Cheng,and Kang(2002)].

Example 2Consider the two-dimensional Laplace equation

subjected to the Dirichlet boundary conditions which are extracted from the exact solutionp(x,y)=3x2y+3xy2−x3−y3[Atluri and Zhu(1998);Zhu,Zhang and Atluri(1998)].

Figure 1 shows the relative error normL2of numerical solutions of Eq.(19),which are obtained respectively by using the local boundary integral equation method(LBIE)[Zhu,Zhang,and Atluri(1998)],the meshless Local Petrov-Galerkin method(MLPG)[Atluri and Zhu(1998)],and the present wavelet method.We see from Figure 1 that the present wavelet solution is very accurate and almost independent of the number of grid pointsN,which is different from those given by LBIE and MLPG whose order of convergence is about 7.5[Atluri and Zhu(1998);Zhu,Zhang,and Atluri(1998)].The reason for this phenomenon may be the fact that the wavelet expansion(1)can exactly characterize the theoretical solution of Eq.(19),since Eq.(1)is a completely accurate representation of the polynomial whose order is below the vanishing moment β=6 of the wavelet function we use[Meyer(1992);Wang(2001);Liu,Zhou,Zhang,and Wang(2014)].And the very slight error of the proposed solution shown in Figure 1 may be caused by rounding errors which are not specially handled in this study.

Figure 1:Relative error norm L2 of numerical solutions of Eq.(19)obtained respectively by using LBIE,MLPG and the present wavelet method.

Example 3Consider the two-dimensional Poisson equation

subjected respectively to the Dirichlet boundary conditions[Barad and Colella(2005)]

and the mixed boundary conditions

The exact solution of this problem isp(x,y)=sin(2πx)sin(2πy).

Table 3:Error L1 and order of convergence R1 for Eq.(20)with boundary conditions(21).

Table 4:Error L∞and order of convergence R∞for Eq.(18)with boundary conditions(21).

Tables 3 and 4 respectively show the comparisons of the mean absolute errorL1and the maximum absolute errorL∞between solutions obtained respectively by using different numerical methods for the two-dimensional Poisson equation(20)with the Dirichlet boundary conditions(21).It can be seen from Tables 3 and 4 that the present wavelet solutions are more accurate than those given by both of the two-order finite volume method(TFVD)and the four-order finite volume method(TFVD)[Barad and Colella(2005)].And from Tables 3 and 4,we also can find that the order of convergenceR∞,1of the proposed wavelet method is about 5,which obviously exceeds the order of convergence of the finite volume method[Barad and Colella(2005)].In Figure 2,we show the relation between the errors and the number of grid pointsNfor the two-dimensional Poisson equation(20)with the mixed boundary conditions(22).It can been seen from Figure 1 that the present wavelet method for solving Poisson problems with complex boundary conditions is also very accurate and efficient,in which the order of convergenceR∞,1≈ 4 and the mean absolute errorL1≈ 2.27× 10−7for the number of grid pointsN=128×128.

Figure 2:Mean absolute error L1 and maximum absolute error L∞of wavelet solutions of Eq.(20)with mixed boundary conditions(22)as a function of the number of grid points in x(y)direction N1/2.

Example 4Consider the three-dimensional Poisson equation with Dirichlet boundary conditions

which has the exact solutionp(x,y,z)=sin(πx)sin(πy)sin(πz)[Zhang(1998)].

Table 5:Maximum absolute error L∞ and order of convergence R∞ for problem(23).

Table 5 lists the maximum absolute errorL∞and order of convergenceR∞of the numerical solutions for the three-dimensional Poisson problem(21)obtained respectively by using the proposed wavelet method and other existing methods.The results listed in Table 5 show clearly that the present wavelet solutions are more accurate than those given respectively by the fourth-order compact scheme(FOS)[Zhang(1998)],the central difference scheme(CDS)[Zhang(1998)],and the Haar wavelet method(HWM)[Shi,Cao,and Chen(2012)].

Figure 3:Mean absolute error L1 and maximum absolute error L∞of wavelet solutions as a function of the number of grid points in x(y,z)direction N1/3.

Example 5Consider the three-dimensional Poisson equation

subjected to the Dirichlet boundary conditions extracted from the exact solutionp(x,y,z)=ex+ey+ez.

Figure 3 shows the mean absolute errorL1and maximum absolute errorL∞of the proposed wavelet solutions as a function of the number of grid pointsN.From Figure 3,we can find out that the present wavelet method has a good accuracy and efficiency for solving the three-dimensional Poisson problem(24),where the order of convergenceR∞,1≈ 4 and the mean absolute errorL1can reach 7.830× 10−9when the number of grid pointsN=32×32×32.

5 Conclusion

In this paper,an approximation scheme for aL2-function defined on a three-dimensional bounded space by combining techniques of boundary extension and Coiflettype wavelet expansion is introduced.Based on such approximation scheme,we proposed a modified wavelet Galerkin method for the solution of Poisson equations with various boundary conditions.By numerically solving the one-,two-and threedimensional Poisson problems,results demonstrate that the proposed wavelet has a much better accuracy and convergence rate than many methods developed so far,and has a good capability in dealing with mixed boundary conditions.

Acknowledgement:This research is supported by grants from the National Natural Science Foundation of China(11421062,11502103),and the Fundamental Research Funds for the Central Universities(lzujbky-2015-178).

Atluri,S.N.;Zhu,T.(1998):A new Meshless Local Petrov-Galerkin(MLPG)approachincomputational mechanics.Computational Mechanics,vol.22,pp.117–127.

Barad,M.;Colella,P.(2005):A fourth-order accurate local refinement method for Poisson’s equation.Journal of Computational Physics,vol.209,pp.1–18.

Bardazzi,A.;Lugni,C.;Antuono,M.;Graziani,G.;Faltinsen,O.M.(2015):Generalized HPC method for the Poisson equation.Journal of Computational Physics,vol.299,pp.630–648.

Chen,R.;Wu,Z.(2006): Applying multiquadric quasi-interpolation to solve Burgers’equation.Applied Mathematics and Computation,vol.172,pp.472–484.

Doha,E.H.(1989):An accurate double Chebyshev spectral approximation for Poisson’s equation.Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae Sectio Computatorica,vol.10,pp.243–276.

Feng,Z.;Sheng,Z.M.(2015):An approach to numerically solving the Poisson equation.Physica Scripta,vol.90,pp.065603.

Gao,Y.;Le,L.H.;Shi,B.C.(2013):Numerical solution of Burgers’equation by lattice Boltzmann method.Applied Mathematics and Computation,vol.219,pp.7685–7692.

Ge,Y.(2010):Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D Poisson equation.Journal of Computational Physics,vol.229,pp.6381–6391.

Gibou,F.;Fedkiw,R.P.;Cheng,L.T.;Kang,M.(2002): A second-orderaccurate symmetric discretization of the Poisson equation on irregular domains.Journal of Computational Physics,vol.176,pp.205–227.

Heise,B.;Kuhn,M.(1996): Parallel solvers for linear and nonlinear exterior magnetic field problems based upon coupled FE/BE formulations.Computing,vol.56,pp.237–258.

Hon,Y.C.;Mao,X.Z.(1998): An efficient numerical scheme for Burgers’equation.Applied Mathematics and Computation,vol.95,pp.37–50.

Howle,V.E.;Vavasis,S.A.(2005):An iterative method for solving complexsymmetric systems arising in electrical power modeling.SIAM Journal on Matrix Analysis and Applications,vol.26,pp.1150–1178.

Jiwari,R.;Mittal,R.C.;Sharma,K.K.(2013):A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’equation.Applied Mathematics and Computation,vol.219,pp.6680–6691.

Kutluay,S.;Esen,A.;Dag,I.(2004):Numerical solutions of the Burgers’equation by the least-squares quadratic B-spline finite element method.Journal of Computational and Applied Mathematics,vol.167,pp.21–33.

Liu,X.J.;Wang,J.Z.;Zhou,Y.H.(2013):A wavelet method for solving nonlinear time-dependent partial differential equations,CMES:Computer Modeling in Engineeringamp;Sciences,vol.94,pp.225–238.

Liu,X.J.;Wang,J.Z.;Zhou,Y.H.(2013): Wavelet solution of a class of two-dimensional nonlinear boundary value problems.CMES:Computer Modeling in Engineeringamp;Sciences,vol.92,pp.493–505.

Liu,X.J.;Zhou,Y.H.;Wang,X.M.;Wang,J.Z.(2013):A wavelet method for solving a class of nonlinear boundary value problems.Communications in Nonlinear Science and Numerical Simulation,vol.18,pp.1939–1948.

Liu,X.J.;Zhou,Y.H.;Zhang,L.;Wang,J.Z.(2014): Wavelet solutions of Burgers’equation with high Reynolds numbers.Science China:Technological Sciences,vol.57,pp.1285–1292.

Meyer,Y.(1992):Wavelets and Operators.Cambridge University Press,Cambridge.

Mittal,R.C.;Gahlaut,S.(1987):High-order finite-differences schemes to solve Poisson’s equation in polar coordinates.IMA Journal of Numerical Analysis,vol.11,pp.261–270.

Ray,S.S.;Gupta,A.K.(2014): An approach with Haar wavelet collocation method for numerical simulations of modified KdV and modified Burgers equations.CMES:Computer Modeling in Engineeringamp;Sciences,vol.103,pp.315–341.

Shi,Z.;Cao,Y.Y.;Chen,Q.J.(2012):Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method.Applied Mathematical Modelling,vol.36,pp.5143–5161.

Vuik,C.;Segal,A.;Meijerink,J.A.(1999):An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients.Journal of Computational Physics,vol.152,pp.385–403.

Wang,J.Z.(2001):Generalized theoryand arithmetic of orthogonal waveletsand applications to researches of mechanics including piezoelectric smart structures.Ph.D.Thesis,Lanzhou University,China.

Wang,X.M.(2014): A wavelet method for solving Bagley-Torvik equation.CMES:Computer Modeling in Engineeringamp;Sciences,vol.102,pp.169–182.

Wordelman,C.J.;Aluru,N.R.;Ravaioli,U.(2000):A meshless method for the numerical solution of the 2-and 3-D semiconductor Poisson equation.CMES:Computer Modeling in Engineeringamp;Sciences,vol.1,pp.121–126.

Xie,H.;Li,D.(2013):A meshless method for Burgers’equation using MQ-RBF and high-order temporal approximation.Applied Mathematical Modelling,vol.37,pp.9215–9222.

Zhang,J.(1998):Fast and high accuracy multigrid solution of the three dimensional Poisson equation.Journal of Computational Physics,vol.143,pp.449–461.

Zhu,C.G.;Wang,R.H.(2009):Numerical solution of Burgers’equation by cubic B-spline quasi-interpolation.Applied Mathematics and Computation,vol.208,pp.260–272.

Zhu,T.;Zhang,J.D.;Atluri,S.N.(1998): A local boundary integral equation(LBIE)method in computational mechanics,and a meshless discretization approach.Computational Mechanics,vol.21,pp.223–235.

1Key Laboratory of Mechanics on Disaster and Environment in Western China,the Ministry of Education,and School of Civil Engineering and Mechanics,Lanzhou University,Lanzhou,Gansu 730000,China.

2Corresponding author.Email:liuxiaojing@lzu.edu.cn.