High Order Accurate Approximation of the First and Pure Second Derivatives of the Laplace Equation on a Rectangle and a Rectangular Parallelepiped

In thisthesis,wediscusstheapproximationofthefirstandpuresecondorderderiva- tivesforthesolutionoftheDirichletproblemforLaplace’sequationonarectangular domain andinarectangularparallelepiped.Inthecasewhenthedomainisarectangle, the boundaryvaluesonthesidesoftherectanglearesupposedtohavesixthderivatives satisfying theHöldercondition.Onthevertices,besidesthecontinuity,thecompat- ibility conditions,whichresultfromtheLaplaceequation,forthesecondandfourth derivativesoftheboundaryfunctions,givenontheadjacentsides,arealsosatisfied. Under theseconditionsauniformapproximationoforder O 􀀀���� h4 (h is thegridsize),is obtained forthesolutionoftheDirichletproblemonasquaregrid,itsfirstandpure second derivatives,byasimpledifferenceschemes. In thecasearectangularparallelepiped,weproposeandjustifydifferenceschemes for thefirstandpuresecondderivativesapproximationofthesolutionoftheDirichlet problem for3DLaplace’sequtation.Theboundaryvaluesonthefacesoftheparal- lelepiped areassumedtohavethesixthderivativessatisfyingtheHöldercondition. Theyarecontinuousontheedges,andtheirsecondandfourthorderderivativessatisfy the compatibilityconditionswhichresultsfromtheLaplaceequation.Itisprovedthat the solutionsoftheproposeddifferenceschemesconvergeuniformlyonthecubicgrid with order O(h4), where h is thegridstep.Forbothcasesnumericalexperimentsare demonstrated tosupporttheanalysismade. Keywords: Finite differencemethod,approximationofderivatives,uniformerror, Laplace equation.