In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary Boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor dep...In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary Boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity,which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor,the velocity and the temperature,whereas the pressure is computed via a postprocessing formula.In addition,an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation.Regarding the discrete problem,we follow the approach employed in a previous work dealing with the Navier-Stokes equations,and couple it with a VEM for the convection-diffusion equation modelling the temperature.More precisely,we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div)and H^(1),respectively,whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H^(1).In this way,we make use of the L^(2)-orthogonal projectors onto suitable polynomial spaces,which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations.On the other hand,in order to manipulate the bilinear form associated to the heat equations,we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor,which represents the thermal conductivity,is variable.Next,the corresponding solvability analysis is performed using again appropriate fixed-point arguments.Further,Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure.The corresponding rates of convergence are also established.Finally,several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.展开更多
We consider a non-standard mixed method for the Stokes problem in Rn,n∈{2,3},with Dirichlet boundary conditions,in which,after using the incompressibility condition to eliminate the pressure,the pseudostress tensor s...We consider a non-standard mixed method for the Stokes problem in Rn,n∈{2,3},with Dirichlet boundary conditions,in which,after using the incompressibility condition to eliminate the pressure,the pseudostress tensor s and the velocity vector u become the only unknowns.Then,we apply the Babuˇska-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations.In particular,we show that Raviart-Thomas elements of order k≥0 for s and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme.In addition,we introduce and analyze an augmented approach for our pseudostress-velocity formulation.The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations,and the Dirichlet boundary condition for the velocity,all of them multiplied by suitable stabilization parameters.We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form,whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces.For instance,Raviart-Thomas elements of order k≥0 for s and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case.Finally,extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature,are provided.展开更多
基金supported by CONICYT-Chile through the project AFB170001 of the PIA Program:Concurso Apoyo a Centros Cientificos y Tecnologicos de Excelencia con Financiamiento Basal,and the Becas-CONICYT Programme for foreign studentsby Centro de Investigacion en Ingenieria Matematica(CI^(2)MA),Universidad de Con-cepcionby Uniyersidad Nacional,Costa Ricea,through the prejeet 0103-18.
文摘In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary Boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity,which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor,the velocity and the temperature,whereas the pressure is computed via a postprocessing formula.In addition,an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation.Regarding the discrete problem,we follow the approach employed in a previous work dealing with the Navier-Stokes equations,and couple it with a VEM for the convection-diffusion equation modelling the temperature.More precisely,we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div)and H^(1),respectively,whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H^(1).In this way,we make use of the L^(2)-orthogonal projectors onto suitable polynomial spaces,which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations.On the other hand,in order to manipulate the bilinear form associated to the heat equations,we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor,which represents the thermal conductivity,is variable.Next,the corresponding solvability analysis is performed using again appropriate fixed-point arguments.Further,Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure.The corresponding rates of convergence are also established.Finally,several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.
文摘We consider a non-standard mixed method for the Stokes problem in Rn,n∈{2,3},with Dirichlet boundary conditions,in which,after using the incompressibility condition to eliminate the pressure,the pseudostress tensor s and the velocity vector u become the only unknowns.Then,we apply the Babuˇska-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations.In particular,we show that Raviart-Thomas elements of order k≥0 for s and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme.In addition,we introduce and analyze an augmented approach for our pseudostress-velocity formulation.The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations,and the Dirichlet boundary condition for the velocity,all of them multiplied by suitable stabilization parameters.We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form,whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces.For instance,Raviart-Thomas elements of order k≥0 for s and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case.Finally,extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature,are provided.
