IN THIS PAPER, WE ARE CONCERNED WITH A MODEL OF THE INDIRECT TRANSMISSION OF AN EPIDEMIC DISEASE BETWEEN TWO SPATIALLY DISTRIBUTED HOST POPULATIONS HAVING NON-COINCIDENT SPATIAL DOMAINS WITH NONLOCAL AND CROSS-DIFFUSION, THE EPIDEMIC DISEASE TRANSMISSION OCCURRING THROUGH A CONTAMINATED ENVIRONMENT. THE MOBILITY OF EACH CLASS IS ASSUMED TO BE INFLUENCED BY THE GRADIENT OF THE OTHER CLASSES. WE ADDRESS THE QUESTIONS OF EXISTENCE OF WEAK SOLUTIONS AND EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTION BY USING, RESPECTIVELY, A REGULARIZATION METHOD AND AN INTERPOLATION RESULT BETWEEN BANACH SPACES. MOREOVER, WE PROPOSE A FINITE VOLUME SCHEME AND PROVED THE WELL-POSEDNESS, NONNEGATIVITY AND CONVERGENCE OF THE DISCRETE SOLUTION. THE CONVERGENCE PROOF IS BASED ON DERIVING A SERIES OF A PRIORI ESTIMATES AND BY USING A GENERAL LP COMPACTNESS CRITERION. FINALLY, THE NUMERICAL SCHEME IS ILLUSTRATED BY SOME EXAMPLES.

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 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 PROPOSE AND ANALYZE AN AUGMENTED MIXED FORMULATION FOR THE TIME-DEPENDENT BRINKMAN-FORCHHEIMER EQUATIONS WRITTEN IN TERMS OF VORTICITY, VELOCITY AND PRESSURE. THE WEAK FORMULATION IS BASED ON THE INTRODUCTION OF SUITABLE LEAST SQUARES TERMS ARISING FROM THE INCOMPRESSIBILITY CONDITION AND THE CONSTITUTIVE EQUATION RELATING THE VORTICITY AND VELOCITY. WE ESTABLISH EXISTENCE AND UNIQUENESS OF A SOLUTION TO THE WEAK FORMULATION, AND DERIVE THE CORRESPONDING STABILITY BOUNDS, EMPLOYING CLASSICAL RESULTS ON NONLINEAR MONOTONE OPERATORS. WE THEN PROPOSE A SEMIDISCRETE CONTINUOUS-IN-TIME APPROXIMATION BASED ON STABLE STOKES ELEMENTS FOR THE VELOCITY AND PRESSURE, AND CONTINUOUS OR DISCONTINUOUS PIECEWISE POLYNOMIAL SPACES FOR THE VORTICITY. IN ADDITION, BY MEANS OF THE BACKWARD EULER TIME DISCRETIZATION, WE INTRODUCE A FULLY DISCRETE FINITE ELEMENT SCHEME. WE PROVE WELL-POSEDNESS AND DERIVE THE STABILITY BOUNDS FOR BOTH SCHEMES, AND ESTABLISH THE CORRESPONDING ERROR ESTIMATES. WE PROVIDE SEVERAL NUMERICAL RESULTS VERIFYING THE THEORETICAL RATES OF CONVERGENCE AND ILLUSTRATING THE PERFORMANCE AND FLEXIBILITY OF THE METHOD FOR A RANGE OF DOMAIN CONFIGURATIONS AND MODEL PARAMETERS.(C) 2022 ELSEVIER B.V. ALL RIGHTS RESERVED.

THIS PAPER DEALS WITH THE NUMERICAL APPROXIMATION OF THE STATIONARY TWO-DIMENSIONAL STOKES EQUATIONS, FORMULATED IN TERMS OF VORTICITY, VELOCITY AND PRESSURE, WITH NON-STANDARD BOUNDARY CONDITIONS. HERE, BY INTRODUCING A GALERKIN LEAST-SQUARES TERM, WE END UP WITH A STABILIZED VARIATIONAL FORMULATION THAT CAN BE RECAST AS A TWOFOLD SADDLE POINT PROBLEM. WE PROPOSE TWO FAMILIES OF MIXED FINITE ELEMENTS TO SOLVE THE DISCRETE PROBLEM, IN THE FIRST FAMILY, THE UNKNOWNS ARE APPROXIMATED BY PIECEWISE CONTINUOUS AND QUADRATIC ELEMENTS, BREZZI?DOUGLAS?MARINI, AND PIECEWISE CONSTANT FINITE ELEMENTS, RESPECTIVELY, WHILE IN THE SECOND FAMILY, THE UNKNOWNS ARE APPROXIMATED BY PIECEWISE LINEAR AND CONTINUOUS, RAVIART?THOMAS, AND PIECEWISE CONSTANT FINITE ELEMENTS, RESPECTIVELY. THE WELLPOSEDNESS OF THE RESULTING CONTINUOUS AND DISCRETE VARIATIONAL PROBLEMS ARE STUDIED EMPLOYING AN EXTENSION OF THE BABU?KA?BREZZI THEORY. WE ESTABLISH A PRIORI ERROR ESTIMATES IN THE NATURAL NORMS, AND WE FINALLY REPORT SOME NUMERICAL EXPERIMENTS ILLUSTRATING THE BEHAVIOR OF THE PROPOSED SCHEMES AND CONFIRMING OUR THEORETICAL FINDINGS ON STRUCTURED AND UNSTRUCTURED MESHES. ADDITIONAL EXAMPLES OF CASES NOT COVERED BY OUR THEORY ARE ALSO PRESENTED.

THIS PAPER DEALS WITH THE ANALYSIS OF A NEW AUGMENTED MIXED FINITE ELEMENT METHOD IN TERMS OF VORTICITY, VELOCITY, AND PRESSURE, FOR THE BRINKMAN PROBLEM WITH NONSTANDARD BOUNDARY CONDITIONS. THE APPROACH IS BASED ON THE INTRODUCTION OF GALERKIN LEAST-SQUARES TERMS ARISING FROM THE CONSTITUTIVE EQUATION RELATING THE AFOREMENTIONED UNKNOWNS AND FROM THE INCOMPRESSIBILITY CONDITION. WE SHOW THAT THE RESULTING AUGMENTED BILINEAR FORM IS CONTINUOUS AND ELLIPTIC, WHICH, THANKS TO THE LAX?MILGRAM THEOREM, AND BESIDES PROVING THE WELL-POSEDNESS OF THE CONTINUOUS FORMULATION, ENSURES THE SOLVABILITY AND STABILITY OF THE GALERKIN SCHEME WITH ANY FINITE ELEMENT SUBSPACE OF THE CONTINUOUS SPACE. IN PARTICULAR, RAVIART?THOMAS ELEMENTS OF ANY ORDER URN:X-WILEY:FLD:MEDIA:FLD4041:FLD4041-MATH-0001 FOR THE VELOCITY FIELD, AND PIECEWISE CONTINUOUS POLYNOMIALS OF DEGREE K + 1 FOR BOTH THE VORTICITY AND THE PRESSURE, CAN BE UTILIZED. A PRIORI ERROR ESTIMATES AND THE CORRESPONDING RATES OF CONVERGENCE ARE ALSO GIVEN HERE. NEXT, WE DERIVE TWO RELIABLE AND EFFICIENT RESIDUAL-BASED A POSTERIORI ERROR ESTIMATORS FOR THIS PROBLEM. THE ELLIPTICITY OF THE BILINEAR FORM TOGETHER WITH THE LOCAL APPROXIMATION PROPERTIES OF THE CLÉMENT INTERPOLATION OPERATOR ARE THE MAIN TOOLS FOR SHOWING THE RELIABILITY. IN TURN, INVERSE INEQUALITIES AND THE LOCALIZATION TECHNIQUE BASED ON TRIANGLE-BUBBLE AND EDGE-BUBBLE FUNCTIONS ARE UTILIZED TO SHOW THE EFFICIENCY. FINALLY, SEVERAL NUMERICAL RESULTS ILLUSTRATING THE GOOD PERFORMANCE OF THE METHOD, CONFIRMING THE PROPERTIES OF THE ESTIMATORS AND SHOWING THE BEHAVIOR OF THE ASSOCIATED ADAPTIVE ALGORITHMS, ARE REPORTED. COPYRIGHT © 2015 JOHN WILEY & SONS, LTD.

WE INTRODUCE A FAMILY OF MIXED METHODS AND DISCONTINUOUS GALERKIN DISCRETISATIONS DESIGNED TO NUMERICALLY SOLVE THE OSEEN EQUATIONS WRITTEN IN TERMS OF VELOCITY, VORTICITY, AND BERNOULLI PRESSURE. THE UNIQUE SOLVABILITY OF THE CONTINUOUS PROBLEM IS ADDRESSED BY INVOKING A GLOBAL INF-SUP PROPERTY IN AN ADEQUATE ABSTRACT SETTING FOR NON-SYMMETRIC SYSTEMS. THE PROPOSED FINITE ELEMENT SCHEMES, WHICH PRODUCE EXACTLY DIVERGENCE-FREE DISCRETE VELOCITIES, ARE SHOWN TO BE WELL-DEFINED AND OPTIMAL CONVERGENCE RATES ARE DERIVED IN SUITABLE NORMS. THIS MIXED FINITE ELEMENT METHOD IS ALSO PRESSURE-ROBUST. IN ADDITION, WE ESTABLISH OPTIMAL RATES OF CONVERGENCE FOR A CLASS OF DISCONTINUOUS GALERKIN SCHEMES, WHICH EMPLOY STABILISATION. A SET OF NUMERICAL EXAMPLES SERVES TO ILLUSTRATE SALIENT FEATURES OF THESE METHODS.

IN THIS WORK, WE HAVE DESIGNED CONFORMING AND NONCONFORMING VIRTUAL ELEMENT METHODS (VEM) TO APPROXIMATE NON-STATIONARY NONLOCAL BIHARMONIC EQUATION ON GENERAL SHAPED DOMAIN. BY EMPLOYING FAEDO?GALERKIN TECHNIQUE, WE HAVE PROVED THE EXISTENCE AND UNIQUENESS OF THE CONTINUOUS WEAK FORMULATION. UPON APPLYING BROUWER?S FIXED POINT THEOREM, THE WELL-POSEDNESS OF THE FULLY DISCRETE SCHEME IS DERIVED. FURTHER, FOLLOWING [J. HUANG AND Y. YU, A MEDIUS ERROR ANALYSIS FOR NONCONFORMING VIRTUAL ELEMENT METHODS FOR POISSON AND BIHARMONIC EQUATIONS, J. COMPUT. APPL. MATH. 386 (2021) 113229], WE HAVE INTRODUCED ENRICHMENT OPERATOR AND DERIVED A PRIORI ERROR ESTIMATES FOR FULLY DISCRETE SCHEMES ON POLYGONAL DOMAINS, NOT NECESSARILY CONVEX. THE PROPOSED ERROR ESTIMATES ARE JUSTIFIED WITH SOME BENCHMARK EXAMPLES.

A VARIATIONAL FORMULATION IS ANALYSED FOR THE OSEEN EQUATIONS WRITTEN IN TERMS OF VORTICITY AND BERNOULLI PRESSURE. THE VELOCITY IS FULLY DECOUPLED USING THE MOMENTUM BALANCE EQUATION, AND IT IS LATER RECOVERED BY A POST-PROCESS. A FINITE ELEMENT METHOD IS ALSO PROPOSED, CONSISTING IN EQUAL-ORDER NÉDÉLEC FINITE ELEMENTS AND PIECEWISE CONTINUOUS POLYNOMIALS FOR THE VORTICITY AND THE BERNOULLI PRESSURE, RESPECTIVELY. THE A PRIORI ERROR ANALYSIS IS CARRIED OUT IN THE L2-NORM FOR VORTICITY, PRESSURE, AND VELOCITY; UNDER A SMALLNESS ASSUMPTION EITHER ON THE CONVECTING VELOCITY, OR ON THE MESH PARAMETER. FURTHERMORE, AN A POSTERIORI ERROR ESTIMATOR IS DESIGNED AND ITS ROBUSTNESS AND EFFICIENCY ARE STUDIED USING WEIGHTED NORMS. FINALLY, A SET OF NUMERICAL EXAMPLES IN 2D AND 3D IS GIVEN, WHERE THE ERROR INDICATOR SERVES TO GUIDE ADAPTIVE MESH REFINEMENT. THESE TESTS ILLUSTRATE THE BEHAVIOUR OF THE NEW FORMULATION IN TYPICAL FLOW CONDITIONS, AND ALSO CONFIRM THE THEORETICAL FINDINGS.

IN THIS BRIEF NOTE, WE INTRODUCE A NON-SYMMETRIC MIXED FINITE ELEMENT FORMULATION FOR BRINKMAN EQUATIONS WRITTEN IN TERMS OF VELOCITY, VORTICITY, AND PRESSURE WITH NON-CONSTANT VISCOSITY. THE ANALYSIS IS PERFORMED BY THE CLASSICAL BABU?KA?BREZZI THEORY, AND WE STATE THAT ANY INF?SUP STABLE FINITE ELEMENT PAIR FOR STOKES APPROXIMATING VELOCITY AND PRESSURE CAN BE COUPLED WITH A GENERIC DISCRETE SPACE OF ARBITRARY ORDER FOR THE VORTICITY. WE ESTABLISH OPTIMAL A PRIORI ERROR ESTIMATES, WHICH ARE FURTHER CONFIRMED THROUGH COMPUTATIONAL EXAMPLES.

WE PROPOSE A NEW LOCKING-FREE FAMILY OF MIXED FINITE ELEMENT AND FINITE VOLUME ELEMENT METHODS FOR THE APPROXIMATION OF LINEAR ELASTOSTATICS, FORMULATED IN TERMS OF DISPLACEMENT, ROTATION VECTOR, AND PRESSURE. THE UNIQUE SOLVABILITY OF THE THREE-FIELD CONTINUOUS FORMULATION, AS WELL AS THE WELL-DEFINITENESS AND STABILITY OF THE PROPOSED GALERKIN AND PETROV?GALERKIN METHODS, IS ESTABLISHED THANKS TO THE BABU?KA?BREZZI THEORY. OPTIMAL A PRIORI ERROR ESTIMATES ARE DERIVED USING NORMS ROBUST WITH RESPECT TO THE LAMÉ CONSTANTS, TURNING THESE NUMERICAL METHODS PARTICULARLY APPEALING FOR NEARLY INCOMPRESSIBLE MATERIALS. WE EXEMPLIFY THE ACCURACY (IN A SUITABLY WEIGHTED NORM), AS WELL THE APPLICABILITY OF THE NEW FORMULATION AND THE MIXED SCHEMES BY CONDUCTING A NUMBER OF COMPUTATIONAL TESTS IN 2D AND 3D, ALSO INCLUDING CASES NOT COVERED BY OUR THEORETICAL ANALYSIS.

THIS PAPER IS DEVOTED TO THE NUMERICAL ANALYSIS OF A FAMILY OF FINITE ELEMENT APPROXIMATIONS FOR THE AXISYMMETRIC, MERIDIAN BRINKMAN EQUATIONS WRITTEN IN TERMS OF THE STREAM-FUNCTION AND VORTICITY. A MIXED FORMULATION IS INTRODUCED INVOLVING APPROPRIATE WEIGHTED SOBOLEV SPACES, WHERE WELL-POSEDNESS IS DERIVED BY MEANS OF THE BABU?KA?BREZZI THEORY. WE INTRODUCE A SUITABLE GALERKIN DISCRETIZATION BASED ON CONTINUOUS PIECEWISE POLYNOMIALS OF DEGREE K?1 FOR ALL THE UNKNOWNS, WHERE ITS SOLVABILITY IS ESTABLISHED USING THE SAME FRAMEWORK AS THE CONTINUOUS PROBLEM. OPTIMAL A PRIORI ERROR ESTIMATES ARE DERIVED, WHICH ARE ROBUST WITH RESPECT TO THE FLUID VISCOSITY, AND VALID ALSO IN THE PURE DARCY LIMIT. A FEW NUMERICAL EXAMPLES ARE PRESENTED TO ILLUSTRATE THE CONVERGENCE AND PERFORMANCE OF THE PROPOSED SCHEMES.

WE ARE INTERESTED IN MODELLING THE INTERACTION OF BACTERIA AND CERTAIN NUTRIENT CONCENTRATION WITHIN A POROUS MEDIUM ADMITTING VISCOUS FLOW. THE GOVERNING EQUATIONS IN PRIMAL-MIXED FORM CONSIST OF AN ADVECTION-REACTION-DIFFUSION SYSTEM REPRESENTING THE BACTERIA-CHEMICAL MASS EXCHANGE, COUPLED TO THE BRINKMAN PROBLEM WRITTEN IN TERMS OF FLUID VORTICITY, VELOCITY AND PRESSURE, AND DESCRIBING THE FLOW PATTERNS DRIVEN BY AN EXTERNAL SOURCE DEPENDING ON THE LOCAL DISTRIBUTION OF THE CHEMICAL SPECIES. A PRIORI STABILITY BOUNDS ARE DERIVED FOR THE UNCOUPLED PROBLEMS, AND THE SOLVABILITY OF THE FULL SYSTEM IS ANALYSED USING A FIXED-POINT APPROACH. WE INTRODUCE A PRIMAL-MIXED FINITE ELEMENT METHOD TO NUMERICALLY SOLVE THE MODEL EQUATIONS, EMPLOYING A PRIMAL SCHEME WITH PIECEWISE LINEAR APPROXIMATION OF THE REACTION-DIFFUSION UNKNOWNS, WHILE THE DISCRETE FLOW PROBLEM USES A MIXED APPROACH BASED ON RAVIART-THOMAS ELEMENTS FOR VELOCITY, NÉDÉLEC ELEMENTS FOR VORTICITY, AND PIECEWISE CONSTANT PRESSURE APPROXIMATIONS. IN PARTICULAR, THIS CHOICE PRODUCES EXACTLY DIVERGENCE-FREE VELOCITY APPROXIMATIONS. WE ESTABLISH EXISTENCE OF DISCRETE SOLUTIONS AND SHOW THEIR CONVERGENCE TO THE WEAK SOLUTION OF THE CONTINUOUS COUPLED PROBLEM. FINALLY, WE REPORT SEVERAL NUMERICAL EXPERIMENTS ILLUSTRATING THE BEHAVIOUR OF THE PROPOSED SCHEME.

IN THIS WORK, WE PROPOSE AND ANALYSE AN AUGMENTED MIXED FINITE ELEMENT METHOD FOR SOLVING THE NAVIER?STOKES EQUATIONS DESCRIBING THE MOTION OF INCOMPRESSIBLE FLUID. THE MODEL IS WRITTEN IN TERMS OF VELOCITY, VORTICITY, AND PRESSURE, AND TAKES INTO ACCOUNT NON-CONSTANT VISCOSITY AND NO-SLIP BOUNDARY CONDITIONS. THE WEAK FORMULATION OF THE METHOD INCLUDES LEAST-SQUARES TERMS THAT ARISE FROM THE CONSTITUTIVE EQUATION AND THE INCOMPRESSIBILITY CONDITION. WE DISCUSS THE THEORETICAL AND PRACTICAL IMPLICATIONS OF USING AUGMENTATION IN DETAIL. ADDITIONALLY, WE USE FIXED?POINT STRATEGIES TO SHOW THE EXISTENCE AND UNIQUENESS OF CONTINUOUS AND DISCRETE SOLUTIONS UNDER THE ASSUMPTION OF SUFFICIENTLY SMALL DATA. THE METHOD IS CONSTRUCTED USING ANY COMPATIBLE FINITE ELEMENT PAIR FOR VELOCITY AND PRESSURE, AS DICTATED BY STOKES INF-SUP STABILITY, WHILE FOR VORTICITY, ANY GENERIC DISCRETE SPACE OF ARBITRARY ORDER CAN BE USED. WE ESTABLISH OPTIMAL A PRIORI ERROR ESTIMATES AND PROVIDE A SET OF NUMERICAL TESTS IN 2D AND 3D TO ILLUSTRATE THE BEHAVIOUR OF THE DISCRETISATIONS AND VERIFY THEIR THEORETICAL CONVERGENCE RATES. OVERALL, THIS METHOD PROVIDES AN EFFICIENT AND ACCURATE SOLUTION FOR SIMULATING FLUID FLOW IN A WIDE RANGE OF SCENARIOS.

IN THIS WORK, WE STUDY A FINITE VOLUME SCHEME FOR A REACTION DIFFUSION SYSTEM WITH CONSTANT AND CROSS DIFFUSION MODELING THE SPREAD OF AN EPIDEMIC DISEASE WITHIN A HOST POPULATION STRUCTURED WITH THREE SUBCLASSES OF INDIVIDUALS (SIR-MODEL). THE MOBILITY IN EACH CLASS IS ASSUMED TO BE INFLUENCED BY THE GRADIENT OF OTHER CLASSES. WE ESTABLISH THE EXISTENCE OF A SOLUTION TO THE FINITE VOLUME SCHEME AND SHOW CONVERGENCE TO A WEAK SOLUTION. THE CONVERGENCE PROOF IS BASED ON DERIVING A SERIES OF A PRIORI ESTIMATES AND USING A GENERAL L P COMPACTNESS CRITERION.

IN THIS WORK, WE STUDY A FINITE VOLUME SCHEME FOR A REACTION DIFFUSION SYSTEM WITH CONSTANT AND CROSS DIFFUSION MODELING THE SPREAD OF AN EPIDEMIC DISEASE WITHIN A HOST POPULATION STRUCTURED WITH THREE SUBCLASSES OF INDIVIDUALS (SIR-MODEL). THE MOBILITY IN EACH CLASS IS ASSUMED TO BE INFLUENCED BY THE GRADIENT OF OTHER CLASSES. WE ESTABLISH THE EXISTENCE OF A SOLUTION TO THE FINITE VOLUME SCHEME AND SHOW CONVERGENCE TO A WEAK SOLUTION. THE CONVERGENCE PROOF IS BASED ON DERIVING A SERIES OF A PRIORI ESTIMATES AND USING A GENERAL L P COMPACTNESS CRITERION.

WE INTRODUCE A NEW FINITE ELEMENT METHOD FOR THE APPROXIMATION OF THE THREE-DIMENSIONAL BRINKMAN PROBLEM FORMULATED IN TERMS OF THE VELOCITY, VORTICITY AND PRESSURE FIELDS. THE PROPOSED STRATEGY EXHIBITS THE ADVANTAGE THAT, AT THE CONTINUOUS LEVEL, A COMPLETE DECOUPLING OF VORTICITY AND PRESSURE CAN BE ESTABLISHED UNDER THE ASSUMPTION OF SUFFICIENT REGULARITY. THE VELOCITY IS THEN OBTAINED AS A SIMPLE POSTPROCESS FROM VORTICITY AND PRESSURE, USING THE MOMENTUM EQUATION. WELL-POSEDNESS FOLLOWS STRAIGHTFORWARDLY BY THE LAX?MILGRAM THEOREM. THE GALERKIN SCHEME IS BASED ON NÉDÉLEC AND PIECEWISE CONTINUOUS FINITE ELEMENTS OF DEGREE K?1 FOR VORTICITY AND PRESSURE, RESPECTIVELY. THE DISCRETE SETTING USES THE VERY SAME IDEAS AS IN THE CONTINUOUS CASE, AND THE ERROR ANALYSIS FOR THE VORTICITY SCHEME IS CARRIED OUT FIRST. AS A BYPRODUCT OF THESE ERROR BOUNDS AND THE PROBLEM DECOUPLING, THE CONVERGENCE RATES FOR THE PRESSURE AND VELOCITY ARE READILY OBTAINED IN THE NATURAL NORMS WITH CONSTANTS INDEPENDENT OF THE VISCOSITY. WE ALSO PRESENT DETAILS ABOUT HOW THE ANALYSIS OF THE METHOD IS MODIFIED FOR AXISYMMETRIC, MERIDIAN BRINKMAN FLOWS; AND MODIFY THE DECOUPLING STRATEGY TO INCORPORATE THE CASE OF DIRICHLET BOUNDARY CONDITIONS FOR THE VELOCITY. A SET OF NUMERICAL EXAMPLES IN TWO AND THREE SPATIAL DIMENSIONS ILLUSTRATE THE ROBUSTNESS AND ACCURACY OF THE FINITE ELEMENT METHOD, AS WELL AS ITS COMPETITIVE COMPUTATIONAL COST COMPARED WITH RECENT FULLY MIXED AND AUGMENTED FORMULATIONS OF INCOMPRESSIBLE FLOWS.

ONE OF THE SIMPLEST DETERMINISTIC MATHEMATICAL MODEL FOR THE SPREAD OF AN EPIDEMIC DISEASE IS THE SO-CALLED SI SYSTEM MADE OF TWO ORDINARY DIFFERENTIAL EQUATIONS. IT EXHIBITS SIMPLE DYNAMICS: A BIFURCATION PARAMETER T0 YIELDING PERSISTENCE OF THE DISEASE WHEN T0>1, ELSE EXTINCTION OCCURS. A NATURAL QUESTION IS WHETHER THIS GENTLE DYNAMIC CAN BE DISTURBED BY SPATIAL DIFFUSION. IT IS STRAIGHTFORWARD TO CHECK IT IS NOT FEASIBLE FOR LINEAR/NONLINEAR DIFFUSIONS. WHEN CROSS DIFFUSION IS INTRODUCED FOR SUITABLE CHOICES OF THE PARAMETER DATA SET THIS PERSISTENT STATE OF THE ODE MODEL SYSTEM BECOMES LINEARLY UNSTABLE FOR THE RESULTING INITIAL AND NO-FLUX BOUNDARY VALUE PROBLEM. ON THE OTHER HAND ?NATURAL? WEAK SOLUTIONS CAN BE DEFINED FOR THIS INITIAL AND NO-FLUX BOUNDARY VALUE PROBLEM AND PROVED TO EXIST PROVIDED NONLINEAR AND CROSS DIFFUSIVITIES SATISFY SOME CONSTRAINTS. THESE CONSTRAINTS ARE NOT FULLY MET FOR THE PARAMETER DATA SET YIELDING INSTABILITY. A REMAINING OPEN QUESTION IS: TO WHICH SOLUTIONS DOES THIS APPLY? PERIODIC BEHAVIORS ARE OBSERVED FOR A SUITABLE RANGE OF CROSS DIFFUSIVITIES.

WE DEVELOP THE A POSTERIORI ERROR ANALYSIS OF THREE MIXED FINITE ELEMENT FORMULATIONS FOR ROTATION-BASED EQUATIONS IN ELASTICITY, POROELASTICITY, AND INTERFACIAL ELASTICITY-POROELASTICITY. THE DISCRETIZATIONS USE H1-CONFORMING FINITE ELEMENTS OF DEGREE K + 1 FOR DISPLACEMENT AND FLUID PRESSURE, AND DISCONTINUOUS PIECEWISE POLYNOMIALS OF DEGREE K FOR ROTATION VECTOR, TOTAL PRESSURE, AND ELASTIC PRESSURE. RESIDUAL-BASED ESTIMATORS ARE CONSTRUCTED, AND UPPER AND LOWER BOUNDS (UP TO DATA OSCILLATIONS) FOR ALL GLOBAL ESTIMATORS ARE RIGOROUSLY DERIVED. THE METHODS ARE ALL ROBUST WITH RESPECT TO THE MODEL PARAMETERS (IN PARTICULAR, THE LAM\'E CONSTANTS); THEY ARE VALID IN 2D AND 3D, AND ALSO FOR ARBITRARY POLYNOMIAL DEGREE K \GEQ 0. THE ERROR BEHAVIOR PREDICTED BY THE THEORETICAL ANALYSIS IS THEN DEMONSTRATED NUMERICALLY ON A SET OF COMPUTATIONAL EXAMPLES INCLUDING DIFFERENT GEOMETRIES ON WHICH WE PERFORM ADAPTIVE MESH REFINEMENT GUIDED BY THE A POSTERIORI ERROR ESTIMATORS.