Artículos científicos

IN THIS PRESENT PAPER, WE PROPOSE AND ANALYZE A -CONFORMING VIRTUAL ELEMENT METHOD TO SOLVE THE SO-CALLED ONE-LAYER STATIONARY QUASI-GEOSTROPHIC EQUATIONS (QGE) WITH APPLICATIONS IN THE LARGE SCALE WIND-DRIVEN OCEAN CIRCULATION, FORMULATED IN TERMS OF THE STREAM-FUNCTION. THIS PROBLEM CORRESPONDS TO A NONLINEAR FOURTH ORDER PARTIAL DIFFERENTIAL EQUATION. THE VIRTUAL SPACE AND THE DISCRETE SCHEME ARE BUILT IN A STRAIGHTFORWARD WAY DUE TO THE FLEXIBILITY OF THE VIRTUAL APPROACH. UNDER THE ASSUMPTION OF SMALL DATA, WE PROVE WELL-POSEDNESS OF THE DISCRETE PROBLEM BY USING A FIXED-POINT STRATEGY AND UNDER STANDARD ASSUMPTIONS ON THE COMPUTATIONAL DOMAIN, WE ESTABLISH ERROR ESTIMATES IN -NORM FOR THE STREAM-FUNCTION. FINALLY, WE REPORT FOUR NUMERICAL EXPERIMENTS THAT ILLUSTRATE THE BEHAVIOR OF THE PROPOSED SCHEME AND CONFIRM OUR THEORETICAL RESULTS ON DIFFERENT FAMILIES OF POLYGONAL MESHES.

IN THIS PAPER, WE PROPOSE AND ANALYZE A C1-VIRTUAL ELEMENT METHOD OF HIGH ORDER TO SOLVE THE BRINKMAN PROBLEM FORMULATED IN TERMS OF THE STREAM FUNCTION. THE VELOCITY IS OBTAINED AS A SIMPLE POST-PROCESS FROM STREAM FUNCTION AND A NOVEL STRATEGY IS WRITTEN TO RECOVER THE FLUID PRESSURE. WE ESTABLISH OPTIMAL A PRIORI ERROR ESTIMATES FOR THE STREAM FUNCTION, VELOCITY AND PRESSURE WITH CONSTANTS INDEPENDENT OF THE VISCOSITY. FINALLY, WE REPORT SOME NUMERICAL TEST ILLUSTRATING THE BEHAVIOR OF THE VIRTUAL SCHEME AND SUPPORTING OUR THEORETICAL RESULTS ON DIFFERENT FAMILIES OF POLYGONAL MESHES.

IN THIS WORK, WE PROPOSE AND ANALYZE A MORLEY-TYPE VIRTUAL ELEMENT METHOD TO APPROXIMATE THE STOMMEL?MUNK MODEL IN STREAM-FUNCTION FORM. THE DISCRETIZATION IS BASED ON THE FULLY NONCONFORMING VIRTUAL ELEMENT APPROACH PRESENTED IN ANTONIETTI ET AL., (2018) AND ZHAO ET AL., (2018). THE ANALYSIS RESTRICTS TO SIMPLY CONNECTED POLYGONAL DOMAINS, NOT NECESSARILY CONVEX. UNDER STANDARD ASSUMPTIONS ON THE COMPUTATIONAL DOMAIN WE DERIVE SOME INVERSE ESTIMATES, NORM EQUIVALENCE AND APPROXIMATION PROPERTIES FOR AN ENRICHING OPERATOR EH DEFINED FROM THE NONCONFORMING SPACE INTO ITS H2-CONFORMING COUNTERPART. WITH THE HELP OF THESE TOOLS WE PROVE OPTIMAL ERROR ESTIMATES FOR THE STREAM-FUNCTION IN BROKEN H2-, H1- AND L2-NORMS UNDER MINIMAL REGULARITY CONDITION ON THE WEAK SOLUTION. EMPLOYING POSTPROCESSING FORMULAS AND ADEQUATE POLYNOMIAL PROJECTIONS WE COMPUTE FROM THE DISCRETE STREAM-FUNCTION FURTHER FIELDS OF INTEREST, SUCH AS: THE VELOCITY AND VORTICITY. MOREOVER, FOR THESE POSTPROCESSED VARIABLES WE ESTABLISH ERROR ESTIMATES. FINALLY, WE REPORT PRACTICAL NUMERICAL EXPERIMENTS ON DIFFERENT FAMILIES OF POLYGONAL MESHES.

IN THIS WORK, A NEW VIRTUAL ELEMENT METHOD (VEM) OF ARBITRARY ORDER K>=2, FOR THE TIME DEPENDENT NAVIER-STOKES EQUATIONS IN STREAM FUNCTION FORM IS PROPOSED AND ANALYSED. USING SUITABLE PROJECTION OPERATORS, THE BILINEAR AND TRILINEAR TERMS ARE DISCRETISED BY ONLY USING THE PROPOSED DEGREES OF FREEDOM ASSOCIATED WITH THE VIRTUAL SPACE. UNDER CERTAIN ASSUMPTIONS ON THE COMPUTATIONAL DOMAIN, ERROR ESTIMATIONS ARE DERIVED AND SHOWN THAT THE METHOD OPTIMALLY CONVERGENT IN BOTH SPACE AND TIME VARIABLES. FINALLY, TO JUSTIFY THE THEORETICAL ANALYSIS, FOUR BENCHMARK EXAMPLES ARE EXAMINED NUMERICALLY.

THE NONCONFORMING MORLEY-TYPE VIRTUAL ELEMENT METHOD FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS FORMULATED IN TERMS OF THE STREAM-FUNCTION ON SIMPLY CONNECTED POLYGONAL DOMAINS (NOT NECESSARILY CONVEX) IS DESIGNED. A RIGOROUS ANALYSIS BY USING A NEW ENRICHING OPERATOR IS DEVELOPED. MORE PRECISELY, BY EMPLOYING SUCH OPERATOR, WE PROVIDE NOVEL DISCRETE SOBOLEV EMBEDDINGS, WHICH ALLOW TO ESTABLISH THE WELL-POSEDNESS OF THE DISCRETE SCHEME AND OBTAIN OPTIMAL ERROR ESTIMATES IN BROKEN H2-, H1- AND 0-NORMS UNDER MINIMAL REGULARITY CONDITION ON THE WEAK SOLUTION. THE VELOCITY AND VORTICITY FIELDS ARE RECOVERED VIA A POSTPROCESSING FORMULAS. FURTHERMORE, A NEW ALGORITHM FOR PRESSURE RECOVERY BASED ON A STOKES COMPLEX SEQUENCE IS PRESENTED. OPTIMAL ERROR ESTIMATES ARE OBTAINED FOR ALL THE POSTPROCESSED VARIABLES. FINALLY, THE THEORETICAL ERROR BOUNDS AND THE GOOD PERFORMANCE OF THE METHOD ARE VALIDATED THROUGH SEVERAL BENCHMARK TESTS.