A posteriori error analysis of an augmented mixed-primal formulation for the stationary Boussinesq model

In an earlier work of us, a new mixed finite element scheme was developed for the Boussinesq model describing natural convection. Our methodology consisted of a fixed-point strategy for the variational problem that resulted after introducing a modified pseudostress tensor and the normal component of the temperature gradient as auxiliary unknowns in the corresponding Navier-Stokes and advection-diffusion equations defining the model, respectively, along with the incorporation of parameterized redundant Galerkin terms. The well-posedness of both the continuous and discrete settings, the convergence of the associated Galerkin scheme, as well as a priori error estimates of optimal order were stated there. In this work we complement the numerical analysis of our aforementioned augmented mixed-primal method by carrying out a corresponding a posteriori error estimation in two and three dimensions. Standard arguments relying on duality techniques, and suitable Helmholtz decompositions are used to derive a global error indicator and to show its reliability. A globally efficiency property with respect to the natural norm is further proved via usual localization techniques of bubble functions. Finally, an adaptive algorithm based on a reliable, fully local and computable a posteriori error estimator induced by the aforementioned one is proposed, and its performance and effectiveness are illustrated through a few numerical examples in two dimensions.     

A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows

In this paper we extend recent results on the a priori and a posteriori error analysis of an augmented mixed finite element method for the linear elasticity problem, to the case of incompressible fluid flows with symmetric stress tensor. Similarly as before, the present approach is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and from the relations defining the pressure in terms of the stress tensor and the rotation in terms of the displacement, all of them multiplied by stabilization parameters. We show that these parameters can be suitably 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. Next, we present a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element scheme. Finally, several numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are reported.     

A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows

In this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The methods are based on a non-standard mixed approach in which the velocity, the pressure, and the pseudostress are the original unknowns. However, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown, which yields a twofold saddle point operator equation as the resulting dual-mixed variational formulation. In addition, a suitable augmented version of the latter showing a single saddle point structure is also considered. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, we show that Raviart–Thomas elements of order k ? 0 for the pseudostress, and piecewise polynomials of degree k for the velocity and its gradient, ensure the well-posedness of the associated Galerkin schemes. Moreover, we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation. Then, we derive a reliable and efficient residual-based a posteriori error estimator for each scheme. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms.     

A priori and a posteriori error analyses of an HDG method for the Brinkman problem

In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the linear Brinkman model of porous media flow in two and three dimensions and with non-homogeneous Dirichlet boundary conditions. We consider a fully-mixed formulation in which the main unknowns are given by the pseudostress, the velocity and the trace of the velocity, whereas the pressure is easily recovered through a simple postprocessing. We show that the corresponding continuous and discrete schemes are well-posed. In particular, we use the projection-based error analysis in order to derive a priori error estimates. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator and showing the expected behavior of the adaptive refinements are presented.     

An augmented fully-mixed finite element method for the stationary Boussinesq problem

In this paper we propose and analyze a new fully-mixed finite element method for the stationary Boussinesq problem. More precisely, we reformulate a previous primal-mixed scheme for the respective model by holding the same modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations for the fluid; and in contrast, we now introduce a new auxiliary vector unknown involving the temperature, its gradient and the velocity for the heat equation. As a consequence, a mixed approach is carried out in heat as well as fluid equation, and differently from the previous scheme, no boundary unknowns are needed, which leads to an improvement of the method from both the theoretical and computational point of view. In fact, the pressure is eliminated and as a result the unknowns are given by the aforementioned auxiliary variables, the velocity and the temperature of the fluid. In addition, for reasons of suitable regularity conditions, the scheme is augmented by using the constitutive and equilibrium equations, and the Dirichlet boundary conditions. Then, the resulting formulation is rewritten as a fixed point problem and its well-posedness is guaranteed by the classical Banach theorem combined with the Lax–Milgram theorem. As for the associated Galerkin scheme, the Brouwer and the Banach fixed point theorems are utilized to establish existence and uniqueness of discrete solution, respectively. In particular, Raviart–Thomas spaces of order k for the auxiliary unknowns and continuous piecewise polynomials of degree \(\le k +1\) for the velocity and the temperature become feasible choices. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence.     

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators.     

Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics

We study the applicability of the discontinuous Petrov–Galerkin (DPG) variational framework for thin-body problems in structural mechanics. Our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate, local computation of the corresponding ‘optimal’ test functions. In the Timoshenko beam problem, the proposed method is shown to provide the best approximation in an energy-type norm which is equivalent to the L₂-norm for all the unknowns, uniformly with respect to the thickness parameter. The same formulation remains valid also for the asymptotic Euler–Bernoulli solution. As another one-dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations. In particular, we derive a special norm for the test space which leads to a robust method in terms of the shell thickness. Finally, we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp-mesh for resolving the boundary layer.     

A FE/BE coupling for the 3D time-dependent eddy current problem. Part II: a posteriori error estimates and adaptive computations

The discontinuous Galerkin method in time for the coupling of conforming finite element and boundary element methods was established in Part I of this paper, where quasi-optimal a priori error estimates are provided. In the second part, we establish a posteriori error estimates and so justify an adaptive space/time-mesh refinement algorithm for the efficient numerical treatment of the time-dependent eddy current problem.     

On the adaptive finite element method of steady-state rolling contact for hyperelasticity in finite deformations

In this paper, we present an adaptive finite element method for steady-state rolling contact in finite deformations along with a residual based a posteriori error estimator for rolling contact problem with Coulomb friction. A general formulation of rolling contact geometry is derived from the point of view of differential geometry. Solvability conditions for the rolling contact problems are discussed. We use Newton's method to solve variational equations derived from a penalty regularization of the finite element approximation of the rolling contact problem. We provide a numerical example to illustrate the method.     

A meshless Galerkin method for Stokes problems using boundary integral equations

A meshless Galerkin scheme for the simulation of two-dimensional incompressible viscous fluid flows in primitive variables is described in this paper. This method combines a boundary integral formulation for the Stokes equation with the moving least-squares (MLS) approximations for construction of trial and test functions for Galerkin approximations. Unlike the domain-type method, this scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns, thus it is especially suitable for the exterior problems. Compared to other meshless methods such as the boundary node method and the element free Galerkin method, in which the MLS is also introduced, boundary conditions do not present any difficulty in using this meshless method. The convergence and error estimates of this approach are presented. Numerical examples are also given to show the efficiency of the method.     

A posteriori error estimates for a mixed-FEM formulation of a non-linear elliptic problem

We consider the numerical solution, via the mixed finite element method, of a non-linear elliptic partial differential equation in divergence form with Dirichlet boundary conditions. Besides the temperature u and the flux $\text{σ}$, we introduce ∇u as a further unknown, which yields a variational formulation with a twofold saddle point structure. We derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem, which does not need any equilibrium property for its solvability. In addition, for specific finite element subspaces of Raviart–Thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems. Our approach does not need the exact finite element solution, but any reasonable approximation of it, such as, for instance, the one obtained with a fully discrete Galerkin scheme. In particular, we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi-linear forms involved. Finally, several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions.     

Adaptive Finite Element Approximation for an Elliptic Optimal Control Problem with Both Pointwise and Integral Control Constraints

In this paper, we study the adaptive finite element approximation for a constrained optimal control problem with both pointwise and integral control constraints. We first obtain the explicit solutions for the variational inequalities both in the continuous and discrete cases. Then a priori error estimates are established, and furthermore equivalent a posteriori error estimators are derived for both the state and the control approximation, which can be used to guide the mesh refinement for an adaptive multi-mesh finite element scheme. The a posteriori error estimators are implemented and tested with promising numerical results.     

A FE/BE coupling for the 3D time-dependent eddy current problem. Part I: a priori error estimates

In this paper the discontinuous Galerkin method in time for the coupling of conforming finite element and boundary element methods is established. We derive quasi-optimal a priori error estimates. Numerical examples prove the new scheme to be useful in practice. A posteriori error control and an adaptive algorithm are studied in Part II of this paper.     

A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff–Love plate

We present energy norm a posteriori error estimates for continuous/discontinuous Galerkin (c/dG) approximations of the Kirchhoff–Love plate problem. The method is based on a continuous displacement field inserted into a symmetric discontinuous Galerkin formulation of the fourth order partial differential equation governing the deflection of a thin plate. We also give explicit formulas for the penalty parameter involved in the formulation.     

Adaptive Finite Element Approximation for a Constrained Optimal Control Problem via Multi-meshes

In this paper, we study adaptive finite element approximation schemes for a constrained optimal control problem. We derive the equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.     

Adaptive Local Postprocessing Finite Element Method for the Navier-Stokes Equations

An adaptive local postprocessing finite element method for the Navier-Stokes equations is presented in this paper. We firstly solve the problem on a relative coarse grid to get a rough approximation. Then, we correct the rough approximation by solving a series of approximate local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, some numerical examples are presented to verify the algorithm.     

Divergence-Free HDG Methods for the Vorticity-Velocity Formulation of the Stokes Problem

We study a hybridizable discontinuous Galerkin method for solving the vorticity-velocity formulation of the Stokes equations in three-space dimensions. We show how to hybridize the method to avoid the construction of the divergence-free approximate velocity spaces, recover an approximation for the pressure and implement the method efficiently. We prove that, when all the unknowns use polynomials of degree k??0, the L ² norm of the errors in the approximate vorticity and pressure converge with order k+1/2 and the error in the approximate velocity converges with order k+1. We achieve this by letting the normal stabilization function go to infinity in the error estimates previously obtained for a hybridizable discontinuous Galerkin method.     

Adaptive finite volume methods for displacement problems in porous media

In this paper we consider adaptive numerical simulation of miscible and immiscible displacement problems in porous media, which are modeled by single and two phase flow equations. Using the IMPES formulation of the two phase flow equation both problems can be treated in the same numerical framework. We discretise the equations by an operator splitting technique where the flow equation is approximated by Raviart-Thomas mixed finite elements and the saturation or concentration equation by vertex centered finite volume methods. Using a posteriori error estimates for both approximation schemes we deduce an adaptive solution algorithm for the system of equations and show the applicability in several examples.     

A Hybrid Staggered Discontinuous Galerkin Method for KdV Equations

A hybrid staggered discontinuous Galerkin method is developed for the Korteweg–de Vries equation. The equation is written into a system of first order equations by introducing auxiliary variables. Two sets of finite element functions are introduced to approximate the solution and the auxiliary variables. The staggered continuity of the two finite element function spaces gives a natural flux condition and trace value on the element boundaries in the derivation of Galerkin approximation. On the other hand, to deal with the third order derivative term an hybridization idea is used and additional flux unknowns are introduced. The auxiliary variables can be eliminated in each element and the resulting algebraic system on the solution and the additional flux unknowns is solved. Stability of the semi discrete form is proven for various boundary conditions. Numerical results present the optimal order of \(L^2\)-errors of the proposed method for a given polynomial order.     

A new adaptive modeling strategy based on optimal control for atomic-to-continuum coupling simulations

A new adaptive method based on an optimal control approach is proposed for adaptive modeling of an atomic-to-continuum coupling method constructed from the Arlequin framework. The coupling method provides for an approximation of the solution to a fully atomic model in which errors may arise due to the misplacement of the overlap region. The objective is thus to determine, a posteriori, the best position of this overlap region. More precisely, the problem is to find the optimal size of the atomic region that one needs to consider in the coupled formulation in order to accurately estimate predefined quantities of interest. In this new adaptive process, the position of the overlapping domain between the two models is conveniently parameterized and iteratively determined by searching for the optimal parameters. The performance of the method is demonstrated on one-dimensional and two-dimensional test problems. In particular, it is observed that the approach yields better convergence with respect to quantities of interest when compared to a classical adaptive approach based on a posteriori estimates of the modeling error.