IN THIS WORK, WE STUDY THE C0-NONCONFORMING VEM FOR THE FOURTH-ORDER EIGENVALUE PROBLEMS MODELING THE VIBRATION AND BUCKLING PROBLEMS OF THIN PLATES WITH CLAMPED BOUNDARY CONDITIONS ON GENERAL SHAPED POLYGONAL DOMAIN, POSSIBLY EVEN NONCONVEX DOMAIN. BY EMPLOYING THE ENRICHING OPERATOR, WE HAVE DERIVED THE CONVERGENCE ANALYSIS IN DISCRETE H2 SEMINORM, AND H1, L2 NORMS FOR BOTH PROBLEMS. WE USE THE BABUSKA-OSBORN SPECTRAL THEORY (BABUSKA AND OSBORN, 1991), TO SHOW THAT THE INTRODUCED SCHEMES PROVIDE WELL APPROXIMATION OF THE SPECTRUM AND PROVE OPTIMAL ORDER OF RATE OF CONVERGENCE FOR EIGENFUNCTIONS AND DOUBLE ORDER OF RATE OF CONVERGENCE FOR EIGENVALUES. FINALLY, NUMERICAL RESULTS ARE PRESENTED TO SHOW THE GOOD PERFORMANCE OF THE METHOD ON DIFFERENT POLYGONAL MESHES.(C) 2022 ELSEVIER B.V. ALL RIGHTS RESERVED.

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 STUDY, WE PRESENT AND ANALYZE A VIRTUAL ELEMENT DISCRETIZATION FOR A NONSELFADJOINT FOURTH-ORDER EIGENVALUE PROBLEM DERIVED FROM THE TRANSMISSION EIGENVALUE PROBLEM. USING SUITABLE PROJECTION OPERATORS, THE SESQUILINEAR FORMS ARE DISCRETIZED BY ONLY USING THE PROPOSED DEGREES OF FREEDOM ASSOCIATED WITH THE VIRTUAL SPACES. UNDER STANDARD ASSUMPTIONS ON THE POLYGONAL MESHES, WE SHOW THAT THE RESULTING SCHEME PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND PROVE AN OPTIMAL-ORDER ERROR ESTIMATE FOR THE EIGENFUNCTIONS AND A DOUBLE ORDER FOR THE EIGENVALUES. FINALLY, WE PRESENT SOME NUMERICAL EXPERIMENTS ILLUSTRATING THE BEHAVIOR OF THE VIRTUAL SCHEME ON DIFFERENT FAMILIES OF 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 PAPER WE ANALYSE A FINITE ELEMENT APPROXIMATION OF THE STOKES EIGENVALUE PROBLEM. WE INTRODUCE A VARIATIONAL FORMULATION RELYING ONLY ON THE PSEUDOSTRESS TENSOR AND PROPOSE A DISCRETIZATION BY MEANS OF THE LOWEST-ORDER BREZZI?DOUGLAS?MARINI MIXED FINITE ELEMENT. HOWEVER, SIMILAR RESULTS HOLD TRUE FOR OTHER H(DIV)-CONFORMING ELEMENTS, LIKE RAVIART?THOMAS ELEMENTS. WE SHOW THAT THE RESULTING SCHEME PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND PROVE OPTIMAL-ORDER ERROR ESTIMATES. FINALLY, WE REPORT SOME NUMERICAL TESTS SUPPORTING OUR THEORETICAL RESULTS.

THE AIM OF THIS PAPER IS TO DEVELOP A FINITE ELEMENT METHOD TO APPROXIMATE THE BUCKLING PROBLEM OF SIMPLY SUPPORTED KIRCHHOFF PLATES SUBJECTED TO GENERAL PLANE STRESS TENSOR. WE INTRODUCE AN AUXILIARY VARIABLE W:=ALFA U (WITH U REPRESENTING THE DISPLACEMENT OF THE PLATE) TO WRITE A VARIATIONAL FORMULATION OF THE SPECTRAL PROBLEM. WE PROPOSE A CONFORMING DISCRETIZATION OF THE PROBLEM, WHERE THE UNKNOWNS ARE APPROXIMATED BY PIECEWISE LINEAR AND CONTINUOUS FINITE ELEMENTS. WE SHOW THAT THE RESULTING SCHEME PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND PROVE OPTIMAL ORDER ERROR ESTIMATES FOR THE EIGENFUNCTIONS AND A DOUBLE ORDER FOR THE EIGENVALUES. FINALLY, WE PRESENT SOME NUMERICAL EXPERIMENTS SUPPORTING OUR THEORETICAL RESULTS.

IN THIS ARTICLE, WE STUDY A MATHEMATICAL MODEL THAT REPRESENTS THE CONCENTRATION POLARIZATION AND OSMOSIS EFFECTS IN A REVERSE OSMOSIS CROSS-FLOW CHANNEL WITH DENSE MEMBRANES AT SOME OF ITS BOUNDARIES. THE FLUID IS MODELED USING THE NAVIER-STOKES EQUATIONS AND THE SOLUTION-DIFFUSION IS USED TO IMPOSE THE MOMENTUM BALANCE ON THE MEMBRANE. THE SCHEME CONSIST OF A CONFORMING FINITE ELEMENT METHOD WITH THE VELOCITY-PRESSURE FORMULATION FOR THE NAVIER-STOKES EQUATIONS, TOGETHER WITH A PRIMAL SCHEME FOR THE CONVECTION-DIFFUSION EQUATIONS. THE NITSCHE

IN THE PRESENT WORK WE PROPOSE AND ANALYZE A FULLY-COUPLED VIRTUAL ELEMENT METHOD OF HIGH ORDER FOR SOLVING THE TWO DIMENSIONAL NONSTATIONARY BOUSSINESQ SYSTEM IN TERMS OF THE STREAM-FUNCTION AND TEMPERATURE FIELDS. THE DISCRETIZATION FOR THE SPATIAL VARIABLES IS BASED ON THE COUPLING C1- AND C0-CONFORMING VIRTUAL ELEMENT APPROACHES, WHILE A BACKWARD EULER SCHEME IS EMPLOYED FOR THE TEMPORAL VARIABLE. WELL-POSEDNESS AND UNCONDITIONAL STABILITY OF THE FULLY-DISCRETE PROBLEM ARE PROVIDED. MOREOVER, ERROR ESTIMATES IN H2- AND H1-NORMS ARE DERIVED FOR THE STREAM-FUNCTION AND TEMPERATURE, RESPECTIVELY. FINALLY, A SET OF BENCHMARK TESTS ARE REPORTED TO CONFIRM THE THEORETICAL ERROR BOUNDS AND ILLUSTRATE THE BEHAVIOR OF THE FULLY-DISCRETE SCHEME.

THE AIM OF THIS PAPER IS TO DEVELOP A FINITE ELEMENT METHOD WHICH ALLOWS COMPUTING THE BUCKLING COEFFICIENTS AND MODES OF A NON-HOMOGENEOUS TIMOSHENKO BEAM. STUDYING THE SPECTRAL PROPERTIES OF A NON-COMPACT OPERATOR, WE SHOW THAT THE RELEVANT BUCKLING COEFFICIENTS CORRESPOND TO ISOLATED EIGENVALUES OF FINITE MULTIPLICITY. OPTIMAL ORDER ERROR ESTIMATES ARE PROVED FOR THE EIGENFUNCTIONS AS WELL AS A DOUBLE ORDER OF CONVERGENCE FOR THE EIGENVALUES USING CLASSICAL ABSTRACT SPECTRAL APPROXIMATION THEORY FOR NON-COMPACT OPERATORS. THESE ESTIMATES ARE VALID INDEPENDENTLY OF THE THICKNESS OF THE BEAM, WHICH LEADS TO THE CONCLUSION THAT THE METHOD IS LOCKING-FREE. NUMERICAL TESTS ARE REPORTED IN ORDER TO ASSESS THE PERFORMANCE OF THE METHOD.

WE PRESENT A MIMETIC APPROXIMATION OF THE REISSNER?MINDLIN PLATE BENDING PROBLEM WHICH USES DEFLECTIONS AND ROTATIONS AS DISCRETE VARIABLES. THE METHOD APPLIES TO VERY GENERAL POLYGONAL MESHES, EVEN WITH NON MATCHING OR NON CONVEX ELEMENTS. WE PROVE LINEAR CONVERGENCE FOR THE METHOD UNIFORMLY IN THE PLATE THICKNESS.

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.

THE PAPER DEALS WITH THE A POSTERIORI ERROR ANALYSIS OF A VIRTUAL ELEMENT METHOD FOR THE STEKLOV EIGENVALUE PROBLEM. THE VIRTUAL ELEMENT METHOD HAS THE ADVANTAGE OF USING GENERAL POLYGONAL MESHES, WHICH ALLOWS IMPLEMENTING EFFICIENTLY MESH REFINEMENT STRATEGIES. WE INTRODUCE A RESIDUAL TYPE A POSTERIORI ERROR ESTIMATOR AND PROVE ITS RELIABILITY AND GLOBAL EFFICIENCY. LOCAL EFFICIENCY ESTIMATES ALSO HOLD, ALTHOUGH IN SOME ELEMENTS THEY INVOLVE BOUNDARY TERMS THAT ARE NOT KNOWN TO BE LOCALLY NEGLIGIBLE. WE USE THE CORRESPONDING ERROR ESTIMATOR TO DRIVE AN ADAPTIVE SCHEME. FINALLY, WE REPORT THE RESULTS OF A COUPLE OF NUMERICAL TESTS, THAT ALLOW US TO ASSESS THE PERFORMANCE OF THIS APPROACH.

THIS PAPER DEALS WITH THE ANALYSIS OF NEW MIXED FINITE ELEMENT METHODS FOR THE BRINKMAN EQUATIONS FORMULATED IN TERMS OF VELOCITY, VORTICITY AND PRESSURE. EMPLOYING THE BABU-KA-BREZZI THEORY, IT IS PROVED THAT THE RESULTING CONTINUOUS AND DISCRETE VARIATIONAL FORMULATIONS ARE WELL-POSED. IN PARTICULAR, WE SHOW THAT RAVIART-THOMAS ELEMENTS OF ORDER K?0 FOR THE APPROXIMATION OF THE VELOCITY FIELD, PIECEWISE CONTINUOUS POLYNOMIALS OF DEGREE K+1 FOR THE VORTICITY, AND PIECEWISE POLYNOMIALS OF DEGREE K FOR THE PRESSURE, YIELD UNIQUE SOLVABILITY OF THE DISCRETE PROBLEM. ON THE OTHER HAND, WE ALSO SHOW THAT FAMILIES OF FINITE ELEMENTS BASED ON BREZZI?DOUGLAS?MARINI ELEMENTS OF ORDER K+1 FOR THE APPROXIMATION OF VELOCITY, PIECEWISE CONTINUOUS POLYNOMIALS OF DEGREE K+2 FOR THE VORTICITY, AND PIECEWISE POLYNOMIALS OF DEGREE K FOR THE PRESSURE ENSURE THE WELL-POSEDNESS OF THE ASSOCIATED GALERKIN SCHEME. WE NOTE THAT THESE FAMILIES PROVIDE EXACTLY DIVERGENCE-FREE APPROXIMATIONS OF THE VELOCITY FIELD. WE ESTABLISH A PRIORI ERROR ESTIMATES IN THE NATURAL NORMS WITH CONSTANTS INDEPENDENT OF THE VISCOSITY AND WE CARRY OUT THE RELIABILITY AND EFFICIENCY ANALYSIS OF A RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR. FINALLY, WE REPORT SEVERAL NUMERICAL EXPERIMENTS ILLUSTRATING THE BEHAVIOUR OF THE PROPOSED SCHEMES AND CONFIRMING OUR THEORETICAL RESULTS ON UNSTRUCTURED MESHES. ADDITIONAL EXAMPLES OF CASES NOT COVERED BY OUR THEORY ARE ALSO PRESENTED.

WE PRESENT A PRIORI AND A POSTERIORI ERROR ANALYSIS OF A VIRTUAL ELEMENT METHOD (VEM) TO APPROXIMATE THE VIBRATION FREQUENCIES AND MODES OF AN ELASTIC SOLID. WE ANALYZE A VARIATIONAL FORMULATION RELYING ONLY ON THE SOLID DISPLACEMENT AND PROPOSE AN $H^1({\OMEGA})$-CONFORMING DISCRETIZATION BY MEANS OF VEM. UNDER STANDARD ASSUMPTIONS ON THE COMPUTATIONAL DOMAIN, WE SHOW THAT THE RESULTING SCHEME PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND PROVE AN OPTIMAL ORDER ERROR ESTIMATE FOR THE EIGENFUNCTIONS AND A DOUBLE ORDER FOR THE EIGENVALUES. SINCE, THE VEM HAS THE ADVANTAGE OF USING GENERAL POLYGONAL MESHES, WHICH ALLOWS IMPLEMENTING EFFICIENTLY MESH REFINEMENT STRATEGIES, WE ALSO INTRODUCE A RESIDUAL-TYPE A POSTERIORI ERROR ESTIMATOR AND PROVE ITS RELIABILITY AND EFFICIENCY. WE USE THE CORRESPONDING ERROR ESTIMATOR TO DRIVE AN ADAPTIVE SCHEME. FINALLY, WE REPORT THE RESULTS OF A COUPLE OF NUMERICAL TESTS THAT ALLOW US TO ASSESS THE PERFORMANCE OF THIS APPROACH.

IN THIS PAPER WE PROPOSE AND ANALYZE A NOVEL STREAM FORMULATION OF THE VIRTUAL ELEMENT METHOD (VEM) FOR THE SOLUTION OF THE STOKES PROBLEM. THE NEW FORMULATION HINGES UPON THE INTRODUCTION OF A SUITABLE STREAM FUNCTION SPACE (CHARACTERIZING THE DIVERGENCE FREE SUBSPACE OF DISCRETE VELOCITIES) AND IT IS EQUIVALENT TO THE VELOCITY-PRESSURE (INF-SUP STABLE) MIMETIC SCHEME PRESENTED IN [L. BEIRA?O DA VEIGA ET AL., J. COMPUT. PHYS., 228 (2009), PP. 7215--7232] (UP TO A SUITABLE REFORMULATION INTO THE VEM FRAMEWORK). BOTH SCHEMES ARE THUS STABLE AND LINEARLY CONVERGENT BUT THE NEW METHOD RESULTS TO BE MORE DESIRABLE AS IT EMPLOYS MUCH LESS DEGREES OF FREEDOM AND IT IS BASED ON A POSITIVE DEFINITE ALGEBRAIC PROBLEM. SEVERAL NUMERICAL EXPERIMENTS ASSESS THE CONVERGENCE PROPERTIES OF THE NEW METHOD AND SHOW ITS COMPUTATIONAL ADVANTAGES WITH RESPECT TO THE MIMETIC ONE.

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.

WE PRESENT A VIRTUAL ELEMENT METHOD (VEM) FOR A NONLOCAL REACTION?DIFFUSION SYSTEM OF THE CARDIAC ELECTRIC FIELD. FOR THIS SYSTEM, WE ANALYZE AN H1-CONFORMING DISCRETIZATION BY MEANS OF VEM THAT CAN MAKE USE OF GENERAL POLYGONAL MESHES. UNDER STANDARD ASSUMPTIONS ON THE COMPUTATIONAL DOMAIN, WE ESTABLISH THE CONVERGENCE OF THE DISCRETE SOLUTION BY CONSIDERING A SERIES OF A PRIORI ESTIMATES AND BY USING A GENERAL LP COMPACTNESS CRITERION. MOREOVER, WE OBTAIN OPTIMAL ORDER SPACE-TIME ERROR ESTIMATES IN THE L2 NORM. FINALLY, WE REPORT SOME NUMERICAL TESTS SUPPORTING THE THEORETICAL RESULTS.

WE PRESENT A VIRTUAL ELEMENT METHOD (VEM) FOR POSSIBLY NONLINEAR ELASTIC AND INELASTIC PROBLEMS, MAINLY FOCUSING ON A SMALL DEFORMATION REGIME. THE NUMERICAL SCHEME IS BASED ON A LOW-ORDER APPROXIMATION OF THE DISPLACEMENT FIELD, AS WELL AS A SUITABLE TREATMENT OF THE DISPLACEMENT GRADIENT. THE PROPOSED METHOD ALLOWS FOR GENERAL POLYGONAL AND POLYHEDRAL MESHES, IT IS EFFICIENT IN TERMS OF NUMBER OF APPLICATIONS OF THE CONSTITUTIVE LAW, AND IT CAN MAKE USE OF ANY STANDARD BLACK-BOX CONSTITUTIVE LAW ALGORITHM. SOME THEORETICAL RESULTS HAVE BEEN DEVELOPED FOR THE ELASTIC CASE. SEVERAL NUMERICAL RESULTS WITHIN THE 2D SETTING ARE PRESENTED, AND A BRIEF DISCUSSION ON THE EXTENSION TO LARGE DEFORMATION PROBLEMS IS INCLUDED.

WE ANALYZE IN THIS PAPER A VIRTUAL ELEMENT APPROXIMATION FOR THE ACOUSTIC VIBRATION PROBLEM. WE CONSIDER A VARIATIONAL FORMULATION RELYING ONLY ON THE FLUID DISPLACEMENT AND PROPOSE A DISCRETIZATION BY MEANS OF H(DIV) VIRTUAL ELEMENTS WITH VANISHING ROTOR. UNDER STANDARD ASSUMPTIONS ON THE MESHES, WE SHOW THAT THE RESULTING SCHEME PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND PROVE OPTIMAL ORDER ERROR ESTIMATES. WITH THIS END, WE PROVE APPROXIMATION PROPERTIES OF THE PROPOSED VIRTUAL ELEMENTS. WE ALSO REPORT SOME NUMERICAL TESTS SUPPORTING OUR THEORETICAL RESULTS.