简介：Inthisarticlewedefineasurfacefiniteelementmethod(SFEM)forthenumericalsolutionofparabolicpartialdifferentialequationsonhypersurfacesFinR~(n+1).ThekeyideaisbasedontheapproximationofFΓbyapolyhedralsurfaceΓ_hconsistingofaunionofsimplices(trianglesforn=2,intervalsforn=1)withverticesonF.AfiniteelementspaceoffunctionsisthendefinedbytakingthecontinuousfunctionsonΓ_hwhicharelinearaffineoneachsimplexofthepolygonalsurface.WeusesurfacegradientstodefineweakformsofellipticoperatorsandnaturallygenerateweakformulationsofellipticandparabolicequationsonΓ.Ourfiniteelementmethodisappliedtoweakformsoftheequations.Thecomputationofthemassandelementstiffnessmatricesaresimpleandstraightforward.Wegiveanexampleoferrorboundsinthecaseofsemi-discretizationinspaceforafourthorderlinearproblem.Numericalexperimentsaredescribedforseverallinearandnonlinearpartialdifferentialequations.Inparticularthepowerofthemethodisdemonstratedbyemployingit,tosolvehighlynonlinearsecondandfourthorderproblemssuchassurfaceAllen-CahnandCahn-Hilliardequationsandsurfacelevelsetequationsforgeodesicmeancuryatureflow.