第27卷第6期2022年12月哈尔滨理工大学学报JOUＲNALOFHAＲBINUNIVEＲSITYOFSCIENCEANDTECHNOLOGYVol.27No.6Dec.2022DFＲ法与DG法解抛物方程和对流扩散方程等价性毕卉，刘磊(哈尔滨理工大学理学院，哈尔滨150080)摘要:研究直接通量重构法(简称DFＲ)和间断Galerkin法(简称DG)求解抛物方程和对流扩散方程的等价性问题。研究过程分两部分:第一部分对于DFＲ法和直接间断Galerkin法求解抛物方程的等价性给出两种证明。第一种证明主要用到K点高斯求积具有2K－1阶代数精度。第二种证明主要用到勒让德多项式、拉登多项式和洛巴托多项式的特殊性质。第二部分对于DFＲ法和局部间断Galerkin法求解对流扩散方程的等价性给出两种证明。主要思想是利用K－1次多项式至多有K－1个不同的零点，从而通过插值法将局部间断Galerkin法所用到的辅助变量直接表达出来。两种方法等价性的证明对于插值法和投影法解偏微分方程的等价性理论做出了进一步完善。关键词:直接间断Galerkin法;局部间断Galerkin法;直接通量重构法;抛物方程;对流扩散方程DOI:10．15938/j．jhust．2022．06．019中图分类号:O241．3文献标志码:A文章编号:1007－2683(2022)06－0152－07EquivalenceBetweenDFＲMethodandDGMethodforSolvingParabolicEquationandConvection-diffusionEquationBIHui，LIULei(SchoolofSciences，HarbinUniversityofScienceandTechnology，Harbin150080，China)Abstract:TheequivalencebetweendirectfluxreconstructionmethodanddiscontinuousGalerkinmethodforsolvingparabolicequationandconvection-diffusionequationisstudied．Theresearchprocessisdividedintotwoparts:thefirstpartgivestwoproofsfortheequivalenceofDFＲmethodanddirectdiscontinuousGalerkinmethodforsolvingparabolicequations．ThefirstproofmainlyusesK-pointGaussquadraturewith2K－1orderalgebraicaccuracy．ThesecondproofmainlyusesthespecialpropertiesofLegendrepolynomials，ＲadaupolynomialandLobattopolynomial．Inthesecondpart，theequivalenceofDFＲmethodandlocaldiscontinuousGalerkinmethodinsolvingconvection-diffusionequationisproved．ThemainideaisthatthepolynomialofdegreeK－1hasatmostK－1differentzeros，sothattheauxiliaryvariablesusedinlocaldiscontinuousGalerkinmethodcanbedirectlyexpressedbyinterpolation．Theproofofequivalencebetweenthetwomethodsimprovesth...
