WE PRESENT THE ANALYSIS OF AN ABSTRACT PARAMETER-DEPENDENT MIXED VARIATIONAL FORMULATION BASED ON VOLTERRA INTEGRALS OF SECOND KIND. ADAPTING THE CLASSIC MIXED THEORY IN THE VOLTERRA EQUATIONS SETTING, WE PROVE THE WELL POSEDNESS OF THE RESULTING SYSTEM. STABILITY AND ERROR ESTIMATES ARE DERIVED, WHERE ALL THE ESTIMATES ARE UNIFORM WITH RESPECT TO THE PERTURBATION PARAMETER. WE PROVIDE APPLICATIONS OF THE DEVELOPED ANALYSIS FOR A VISCOELASTIC TIMOSHENKO BEAM AND REPORT NUMERICAL TESTS FOR THIS PROBLEM. WE ALSO COMMENT, NUMERICALLY, THE PERFORMANCE OF A VISCOELASTIC REISSNER?MINDLIN PLATE.

IN THIS PAPER WE ANALYZE A POSTERIORI ERROR ESTIMATES FOR A MIXED FORMULATION OF THE LINEAR ELASTICITY EIGENVALUE PROBLEM. A POSTERIORI ESTIMATORS FOR THE NEARLY AND PERFECTLY COMPRESSIBLE ELASTICITY SPECTRAL PROBLEMS ARE PROPOSED. WITH A POST-PROCESS ARGUMENT, WE ARE ABLE TO PROVE RELIABILITY AND EFFICIENCY FOR THE PROPOSED ESTIMATORS. THE NUMERICAL METHOD IS BASED IN RAVIART-THOMAS ELEMENTS TO APPROXIMATE THE PSEUDOSTRESS AND PIECEWISE POLYNOMIALS FOR THE DISPLACEMENT. WE ILLUSTRATE OUR RESULTS WITH NUMERICAL TESTS IN TWO AND THREE DIMENSIONS.

IN TWO AND THREE DIMENSIONS, WE DESIGN AND ANALYZE A POSTERIORI ERROR ESTIMATORS FOR THE MIXED STOKES EIGENVALUE PROBLEM. THE UNKNOWNS ON THIS MIXED FORMULATION ARE THE PSEUDOTRESS, VELOCITY AND PRESSURE. WITH A LOWEST ORDER MIXED FINITE ELEMENT SCHEME, TOGETHER WITH A POSTPROCRESSING TECHNIQUE, WE PROVE THAT THE PROPOSED ESTIMATOR IS RELIABLE AND EFFICIENT. WE ILLUSTRATE THE RESULTS WITH SEVERAL NUMERICAL TESTS IN TWO AND THREE DIMENSIONS IN ORDER TO ASSESS THE PERFORMANCE OF THE ESTIMATOR.

WE DESIGN AND ANALYZE A POSTERIORI ERROR ESTIMATORS FOR THE STOKES SYSTEM WITH SINGULAR SOURCES IN SUITABLE SPACES. WE CONSIDER CLASSICAL LOW-ORDER INF-SUP STABLE AND STABILIZED FINITE ELEMENT DISCRETIZATIONS. WE PROVE, IN TWO AND THREE DIMENSIONAL LIPSCHITZ, BUT NOT NECESSARILY CONVEX POLYTOPAL DOMAINS, THAT THE DEVISED ERROR ESTIMATORS ARE RELIABLE AND LOCALLY EFFICIENT. ON THE BASIS OF THE DEVISED ERROR ESTIMATORS, WE DESIGN A SIMPLE ADAPTIVE STRATEGY THAT YIELDS OPTIMAL EXPERIMENTAL RATES OF CONVERGENCE FOR THE NUMERICAL EXAMPLES THAT WE PERFORM.

THE AIM OF THE PRESENT WORK IS TO DERIVE ERROR ESTIMATES FOR THE LAPLACE EIGENVALUE PROBLEM IN MIXED FORM, IMPLEMENTING A VIRTUAL ELEMENT METHOD. WITH THE AID OF THE THEORY FOR NON-COMPACT OPERATORS, WE PROVE THAT THE PROPOSED METHOD IS SPURIOUS FREE AND CONVERGENT. OPTIMAL ORDER OF CONVERGENCE FOR THE EIGENVALUES AND EIGENFUNCTIONS ARE DERIVED. FINALLY, WE REPORT NUMERICAL TESTS TO CONFIRM THE THEORETICAL RESULTS TOGETHER WITH A RIGOROUS COMPUTATIONAL ANALYSIS OF THE EFFECTS OF THE STABILIZATION PARAMETER, INHERENT FOR THE VIRTUAL ELEMENT METHODS, IN THE COMPUTATION OF THE SPECTRUM.

IN THIS PAPER WE ANALYZE A VIRTUAL ELEMENT METHOD (VEM) FOR A PSEUDOSTRESS FORMULATION OF THE STOKES EIGENVALUE PROBLEM. THIS FORMULATION ALLOWS TO ELIMINATE THE VELOCITY AND THE PRESSURE, LEADING TO AN ELLIPTIC FORMULATION WHERE THE ONLY UNKNOWN IS THE PSEUDOSTRESS TENSOR. ADAPTING THE NON-COMPACT OPERATOR THEORY, WE PROVE THAT OUR METHOD PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND IS SPURIOUS FREE. WE DERIVE OPTIMAL A PRIORI ERROR ESTIMATES, WHICH WE CONFIRM WITH SOME NUMERICAL TESTS. ALSO WE PRESENT A COMPUTATIONAL SPURIOUS ANALYSIS OF THE PROPOSED METHOD.

IN THIS PAPER WE ANALYZE A VIRTUAL ELEMENT METHOD FOR THE TWO DIMENSIONAL ELASTICITY PROBLEM ALLOWING SMALL EDGES. WITH THIS APPROACH, THE CLASSIC ASSUMPTIONS ON THE GEOMETRICAL FEATURES OF THE POLYGONAL MESHES CAN BE RELAXED. IN PARTICULAR, WE CONSIDER ONLY STAR-SHAPED POLYGONS FOR THE MESHES. SUITABLE ERROR ESTIMATES ARE PRESENTED, WHERE A RIGOROUS ANALYSIS ON THE INFLUENCE OF THE LAMÉ CONSTANTS IN EACH ESTIMATE IS PRESENTED. WE REPORT NUMERICAL TESTS TO ASSESS THE PERFORMANCE OF THE METHOD.

THE AIM OF THIS PAPER IS TO ANALYZE A VIRTUAL ELEMENT METHOD FOR THE TWO DIMENSIONAL ELASTICITY SPECTRAL PROBLEM, WHERE THE POLYGONAL MESHES ALLOW FOR THE PRESENCE OF SMALL EDGES. UNDER THIS APPROACH AND WITH THE AID OF THE THEORY OF COMPACT OPERATORS, WE PROVE CONVERGENCE FOR THE PROPOSED VEM AND ERROR ESTIMATES. WE REPORT A SERIES OF NUMERICAL TESTS IN ORDER TO ASSESS THE PERFORMANCE OF THE METHOD WHERE WE ANALYZE THE EFFECTS OF THE POISSON RATIO ON THE COMPUTATION OF THE ORDER OF CONVERGENCE, TOGETHER WITH THE EFFECTS OF THE STABILIZATION TERM ON THE ARISING OF SPURIOUS EIGENVALUES.

THE AIM OF THIS PAPER IS TO ANALYZE THE INFLUENCE OF SMALL EDGES IN THE COMPUTATION OF THE SPECTRUM OF THE STEKLOV EIGENVALUE PROBLEM BY A LOWEST ORDER VIRTUAL ELEMENT METHOD. UNDER WEAKER ASSUMPTIONS ON THE POLYGONAL MESHES, WHICH CAN PERMIT ARBITRARILY SMALL EDGES WITH RESPECT TO THE ELEMENT DIAMETER, WE SHOW THAT THE SCHEME PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND PROVE OPTIMAL ERROR ESTIMATES FOR THE EIGENFUNCTIONS AND A DOUBLE ORDER FOR THE EIGENVALUES. FINALLY, WE REPORT SOME NUMERICAL TESTS SUPPORTING THE THEORETICAL RESULTS.

IN THIS PAPER WE ANALYZE A FINITE ELEMENT METHOD FOR SOLVING A QUADRATIC EIGENVALUE PROBLEM DERIVED FROM THE ACOUSTIC VIBRATION PROBLEM FOR A HETEROGENEOUS DISSIPATIVE FLUID. THE PROBLEM IS SHOWN TO BE EQUIVALENT TO THE SPECTRAL PROBLEM FOR A NONCOMPACT OPERATOR AND ATHOROUGH SPECTRAL CHARACTERIZATION IS GIVEN. THE NUMERICAL DISCRETIZATION OF THE PROBLEM IS BASED ON RAVIART-THOMAS FINITE ELEMENTS. THE METHOD IS PROVED TO BE FREE OF SPURIOUS MODES AND TO CONVERGE WITH OPTIMAL ORDER. FINALLY, WE REPORT NUMERICAL TESTS WHICH ALLOW US TO ASSESS THE PERFORMANCE OF THE METHOD.

WE ANALYZE, IN TWO DIMENSIONS, AN OPTIMAL CONTROL PROBLEM FOR THE NAVIER?STOKES EQUATIONS WHERE THE CONTROL VARIABLE CORRESPONDS TO THE AMPLITUDE OF FORCES MODELED AS POINT SOURCES; CONTROL CONSTRAINTS ARE ALSO CONSIDERED. THIS PARTICULAR SETTING LEADS TO SOLUTIONS TO THE STATE EQUATION EXHIBITING REDUCED REGULARITY PROPERTIES. WE OPERATE UNDER THE FRAMEWORK OF MUCKENHOUPT WEIGHTS, MUCKENHOUPT-WEIGHTED SOBOLEV SPACES, AND THE CORRESPONDING WEIGHTED NORM INEQUALITIES AND DERIVE THE EXISTENCE OF OPTIMAL SOLUTIONS AND FIRST- AND, NECESSARY AND SUFFICIENT, SECOND-ORDER OPTIMALITY CONDITIONS.

THE ACKNOWLEDGEMENTS SECTION WAS MISSING FROM THIS ARTICLE AND SHOULD HAVE READ: ACKNOWLEDGEMENTS THE FIRST AUTHOR HAS BEEN PARTIALLY SUPPORTED BY DICREA THROUGH PROJECT 2120173 GI/C UNIVERSIDAD DEL BÍO-BÍO AND ANID-CHILE THROUGH FONDECYT PROJECT 11200529, CHILE. THE SECOND AUTHOR WAS PARTIALLY SUPPORTED BY DICREA THROUGH PROJECT 2120173 GI/C UNIVERSIDAD DEL BÍO-BÍO, BY THE NATIONAL AGENCY FOR RESEARCH AND DEVELOPMENT, ANID-CHILE THROUGH FONDECYT PROJECT 1220881, BY PROJECT ANILLO OF COMPUTATIONAL MATHEMATICS FOR DESALINATION PROCESSES ACT210087, AND BY PROJECT CENTRO DE MODELAMIENTO MATEMÁTICO (CMM), FB210005, BASAL FUNDS FOR CENTERS OF EXCELLENCE. THE THIRD AUTHOR WAS SUPPORTED BY THROUGH PROJECT R02/21 UNIVERSIDAD DE LOS LAGOS. THE ORIGINAL ARTICLE HAS BEEN CORRECTED.

IN TWO AND THREE DIMENSION WE ANALYZE DISCONTINUOUS GALERKIN METHODS (DG) FOR THE ACOUSTIC VIBRATION PROBLEM. THROUGH ALL OUR STUDY WE CONSIDER AN INVISCID FLUID, LEADING TO A LINEAR EIGENVALUE PROBLEM. THE ACOUSTIC PROBLEM IS WRITTEN, IN FIRST PLACE, IN TERMS OF THE DISPLACEMENT. UNDER THE APPROACH OF THE NON-COMPACT OPERATORS THEORY, WE PROVE CONVERGENCE AND ERROR ESTIMATES FOR THE METHOD WHEN THE DISPLACEMENT FORMULATION IS CONSIDERED. WE ANALYZE THE INFLUENCE OF THE STABILIZATION PARAMETER ON THE COMPUTATION OF THE SPECTRUM, WHERE SPURIOUS EIGENMODES ARISE WHEN THIS PARAMETER IS NOT CORRECTLY CHOSEN. ALTERNATIVELY WE PRESENT THE FORMULATION DEPENDING ONLY ON THE PRESSURE, COMPARING THE PERFORMANCE OF THE DG METHODS WITH THE PURE DISPLACEMENT FORMULATION. COMPUTATIONALLY, WE STUDY THE INFLUENCE OF THE STABILIZATION PARAMETER ON THE ARISING OF SPURIOUS EIGENVALUES WHEN THE SPECTRUM IS COMPUTED. ALSO, WE REPORT TESTS IN TWO AND THREE DIMENSIONS WHERE CONVERGENCE RATES ARE REPORTED, TOGETHER WITH A COMPARISON BETWEEN THE DISPLACEMENT AND PRESSURE FORMULATIONS FOR THE PROPOSED DG METHODS.

IN THIS PAPER WE ANALYZE A MIXED DISPLACEMENT-PSEUDOSTRESS FORMULATION FOR THE ELASTICITY EIGENVALUE PROBLEM. WE PROPOSE A FINITE ELEMENT METHOD TO APPROXIMATE THE PSEUDOSTRESS TENSOR WITH RAVIART-THOMAS ELEMENTS AND THE DISPLACEMENT WITH PIECEWISE POLYNOMIALS. WITH THE AID OF THE CLASSIC THEORY FOR COMPACT OPERATORS, WE PROVE THAT OUR METHOD IS CONVERGENT AND DOES NOT INTRODUCE SPURIOUS MODES. ALSO, WE OBTAIN ERROR ESTIMATES FOR THE PROPOSED METHOD. FINALLY, WE REPORT SOME NUMERICAL TESTS SUPPORTING THE THEORETICAL RESULTS.

THE AIM OF THIS PAPER IS TO ANALYZE A MIXED FORMULATION FOR THE TWO DIMENSIONAL STOKES EIGENVALUE PROBLEM WHERE THE UNKNOWNS ARE THE STRESS AND THE VELOCITY, WHEREAS THE PRESSURE CAN BE RECOVERED WITH A SIMPLE POSTPROCESS OF THE STRESS. THE STRESS TENSOR IS WRITTEN IN TERMS OF THE VORTICITY OF THE FLUID, LEADING TO AN ALTERNATIVE MIXED FORMULATION THAT INCORPORATES THIS PHYSICAL FEATURE. WE PROPOSE A MIXED NUMERICAL METHOD WHERE THE STRESS IS APPROXIMATED WITH SUITABLE NÉDELEC FINITE ELEMENTS, WHEREAS THE VELOCITY IS APPROXIMATED WITH PIECEWISE POLYNOMIALS OF DEGREE . WITH THE AID OF THE COMPACT OPERATORS THEORY WE DERIVE CONVERGENCE OF THE METHOD AND SPECTRAL CORRECTNESS. MOREOVER, WE PROPOSE A RELIABLE AND EFFICIENT A POSTERIORI ERROR ESTIMATOR FOR OUR SPECTRAL PROBLEM IN ORDER TO PROVIDE AN ADAPTIVE STRATEGY TO ACHIEVE THE OPTIMAL ORDER OF CONVERGENCE FOR NON SUFFICIENT SMOOTH EIGENFUNCTIONS. WE REPORT NUMERICAL TESTS WHERE THE SPECTRUM IS COMPUTED, TOGETHER WITH A COMPUTATIONAL ANALYSIS FOR THE PROPOSED ESTIMATOR. IN ADDITION, WE USE THE CORRESPONDING ERROR ESTIMATOR TO DRIVE AN ADAPTIVE SCHEME, AND WE REPORT THE RESULTS OF A NUMERICAL TEST, THAT ALLOW US TO ASSESS THE PERFORMANCE OF THIS APPROACH.

WE ANALYZE, ON TWO DIMENSIONAL POLYGONAL DOMAINS, CLASSICAL LOW?ORDER INF-SUP STABLE FINITE ELEMENT APPROXIMATIONS OF THE STATIONARY NAVIER?STOKES EQUATIONS WITH SINGULAR SOURCES. WE OPERATE UNDER THE ASSUMPTIONS THAT THE CONTINUOUS AND DISCRETE SOLUTIONS ARE SUFFICIENTLY SMALL. WE PERFORM AN A PRIORI ERROR ANALYSIS ON CONVEX DOMAINS. ON LIPSCHITZ, BUT NOT NECESSARILY CONVEX, POLYGONAL DOMAINS, WE DESIGN AN A POSTERIORI ERROR ESTIMATOR AND PROVE ITS GLOBAL RELIABILITY. WE ALSO EXPLORE EFFICIENCY ESTIMATES. WE ILLUSTRATE THE THEORY WITH NUMERICAL TESTS.

WE PROPOSE AND ANALYZE A FINITE ELEMENT METHOD FOR THE OSEEN EIGENVALUE PROBLEM. THIS PROBLEM IS AN EXTENSION OF THE STOKES EIGENVALUE PROBLEM, WHERE THE PRESENCE OF THE CONVECTIVE TERM LEADS TO A NON-SYMMETRIC PROBLEM AND HENCE, TO COMPLEX EIGENVALUES AND EIGENFUNCTIONS. WITH THE AID OF THE COMPACT OPERATORS THEORY, WE PROVE THAT FOR INF-SUP STABLE FINITE ELEMENTS THE CONVERGENCE HOLDS AND HENCE, ERROR ESTIMATES FOR THE EIGENVALUES AND EIGENFUNCTIONS ARE DERIVED. WE ALSO PROPOSE AN A POSTERIORI ERROR ESTIMATOR WHICH RESULTS TO BE RELIABLE AND EFFICIENT. WE REPORT A SERIES OF NUMERICAL TESTS IN TWO AND THREE DIMENSION IN ORDER TO ASSESS THE PERFORMANCE OF THE METHOD AND THE PROPOSED ESTIMATOR.

IN THIS PAPER, WE ANALYZE DISCONTINUOUS GALERKIN METHODS BASED IN THE INTERIOR PENALTY METHOD IN ORDER TO APPROXIMATE THE EIGENVALUES AND EIGENFUNCTIONS OF THE STOKES EIGENVALUE PROBLEM. THE CONSIDERED METHODS IN THIS WORK ARE BASED IN DISCONTINUOUS POLYNOMIALS APPROXIMATIONS FOR THE VELOCITY FIELD AND THE PRESSURE FLUCTUATION IN TWO AND THREE DIMENSIONS. THE METHODS UNDER CONSIDERATION ARE SYMMETRIC AND NONSYMMETRIC, LEADING TO VARIATIONS ON THE ASSOCIATED MATRICES AND, HENCE, ON THE COMPUTATION OF THE EIGENVALUES AND EIGENFUNCTIONS WHERE REAL AND COMPLEX RESULTS MAY APPEAR, DEPENDING ON THE CHOICE OF THE METHOD. WE DERIVE A CONVERGENCE RESULT AND ERROR ESTIMATES FOR THE PROPOSED METHODS, TOGETHER WITH A RIGOROUS COMPUTATIONAL ANALYSIS OF THE EFFECTS OF THE STABILIZATION PARAMETER IN THE APPEARANCE OF SPURIOUS MODES WHEN THE SPECTRUM IS COMPUTED, WHEN SYMMETRIC AND NONSYMMETRIC METHODS ARE PERFORMED.

IN THIS PAPER WE STUDY A FINITE ELEMENT FORMULATION FOR TIMOSHENKO BEAMS. IT IS KNOWN THAT STANDARD FINITE ELEMENTS APPLIED TO THIS MODEL LEAD TO WRONG RESULTS WHEN THE THICKNESS OF THE BEAM IS SMALL. HERE, WE CONSIDER A MIXED FORMULATION IN TERMS OF THE TRANSVERSE DISPLACEMENT, ROTATION, SHEAR STRESS AND BENDING MOMENT. BY USING THE CLASSICAL BABU?KA?BREZZI THEORY, IT IS PROVED THAT THE RESULTING VARIATIONAL FORMULATION IS WELL POSED. WE DISCRETIZE IT BY CONTINUOUS PIECEWISE LINEAR FINITE ELEMENTS FOR THE SHEAR STRESS AND BENDING MOMENT, AND DISCONTINUOUS PIECEWISE CONSTANT FINITE ELEMENTS FOR THE DISPLACEMENT AND ROTATION. WE PROVE AN OPTIMAL (LINEAR) ORDER OF CONVERGENCE IN TERMS OF THE MESH SIZE FOR THE NATURAL NORMS AND A DOUBLE ORDER (QUADRATIC) IN -NORMS FOR THE SHEAR STRESS AND THE BENDING MOMENT. THESE ESTIMATES INVOLVE CONSTANTS AND NORMS OF THE SOLUTION THAT ARE PROVED TO BE BOUNDED INDEPENDENTLY OF THE BEAM THICKNESS, WHICH ENSURES THE LOCKING-FREE CHARACTER OF THE METHOD. NUMERICAL TESTS ARE REPORTED IN ORDER TO SUPPORT OUR THEORETICAL RESULTS.

WE INTRODUCE A DISCONTINUOUS GALERKIN METHOD FOR THE MIXED FORMULATION OF THE ELASTICITY EIGENPROBLEM WITH REDUCED SYMMETRY. THE ANALYSIS OF THE RESULTING DISCRETE EIGENPROBLEM DOES NOT FIT IN THE STANDARD SPECTRAL APPROXIMATION FRAMEWORK SINCE THE UNDERLYING SOURCE OPERATOR IS NOT COMPACT AND THE SCHEME IS NONCONFORMING. WE SHOW THAT THE PROPOSED SCHEME PROVIDES A CORRECT APPROXIMATION OF THE SPECTRUM AND PROVE ASYMPTOTIC ERROR ESTIMATES FOR THE EIGENVALUES AND THE EIGENFUNCTIONS. FINALLY, WE PROVIDE SEVERAL NUMERICAL TESTS TO ILLUSTRATE THE PERFORMANCE OF THE METHOD AND CONFIRM THE THEORETICAL RESULTS.