Spectral Method/B Spline basis function
 

HI ALL, I AM USING SPECTRAL METHOD WITH B SPLINE AS THE BASIS FUNCTION FOR SOLVING THE N-S EQUATION(STEADY STATE LID DRIVEN CAVITY FLOW PROBLEM IN 2D). I AM USING NEWTON-RAPHSON METHOD FOR SOLVING SYSTEM OF NON LINEAR EQUATIONS. THE PROBLEM IS THAT JACOBAIN MATRIX OBTAINED AT EACH STEP OF THE N-R ITERATION IS VERY ILL CONDITIONED. FOR THE VERY FIRST ITERATION, IF I REPLACE ONLY ONE OF THE CONTINUITY EQUATION BY RANDOM EQUATIONS THEN CONDITION NUMBER IMPROVES CONSIDERABLY..IT MEANS THAT ONE OF THE CONTINUITY EQUATION IS LINEAR COMBINATION OF SOME OTHER EQUATION(??). HOWEVER..THE STORY IS ENTIRELY DIFFEENT WITH THE BIGGER SIZE MATRIX(REFINED MESH). AS I MAKE THE MESH MORE AND MORE REFINED, THE CONDITION NUMBER OF THE JACOBIAN BECOMES WORSE AND WORSE. I AM USING STAGGERED KNOT SEQUENCE FOR THE CALCULATION OF U AND P B -SPLINES. AND EVEYTHINGS SEEMS TO BE FINE IN MY CODE SO FAR(TYPO ERRORS ETC ETC..) ANY SUGGESTION..OR COMMENTS ABOUT THE ORIGIN OF THE (ALMOST) SINGULARITY OF THE JACOBIAN MATRIX.. AND REMEDIES TO REMOVE THE SAME. THANX IN ADVANCE,