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

In this paper we undertake an a posteriori error analysis along with its adaptive computation of a new augmented fully-mixed finite element method that we have recently proposed to numerically simulate heat driven flows in the Boussinesq approximation setting. Our approach incorporates as additional unknowns a modified pseudostress tensor field and an auxiliary vector field in the fluid and heat equations, respectively, which possibilitates the elimination of the pressure. This unknown, however, can be easily recovered by a postprocessing formula. In turn, redundant Galerkin terms are included into the weak formulation to ensure well-posedness. In this way, the resulting variational formulation is a four-field augmented scheme, whose Galerkin discretization allows a Raviart–Thomas approximation for the auxiliary unknowns and a Lagrange approximation for the velocity and the temperature. In the present work, we propose a reliable and efficient, fully-local and computable, residual-based a posteriori error estimator in two and three dimensions for the aforementioned method. Standard arguments based on duality techniques, stable Helmholtz decompositions, and well-known results from previous works, are the main underlying tools used in our methodology. Several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm.     

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 variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity

This paper presents an error estimation framework for a mixed displacement–pressure finite element method for nearly incompressible elasticity. The proposed method is based on Variational Multiscale (VMS) concepts, wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh. Variational projection of fine scales onto the coarse-scale space via variational embedding of the fine-scale solution into the coarse-scale formulation leads to the stabilized method with two major attributes: first, it is free of volumetric locking and, second, it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields. This VMS-based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error. Specifically, two error estimators are explored. The first method employs an element-based strategy and a representation of error via a fine-scale error equation defined over element interiors which is evaluated by a direct post-solution evaluation. This quantity when combined with the global pollution error results in a simple explicit error estimator. The second method involves solving the fine-scale error equation through localization to overlapping patches spread across the domain, thereby leading to an implicit calculation of the local error. This implicit calculation when combined with the global pollution error results in an implicit error estimator. The performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes. The sharpness and robustness of the estimators is shown to be consistent across the test cases employed.     

A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem

In this paper we propose a new a posteriori error estimator for a boundary element solution related to a Dirichlet problem with a second order elliptic partial differential operator. The method is based on an approximate solution of a boundary integral equation of the second kind by a Neumann series to estimate the error of a previously computed boundary element solution. For this one may use an arbitrary boundary element method, for example, a Galerkin, collocation or qualocation scheme, to solve an appropriate boundary integral equation. Due to the approximate solution of the error equation the proposed estimator provides high accuracy. A numerical example supports the theoretical results. Received: June 1999 / Accepted: September 1999     

A flux-free a posteriori error estimator for the incompressible Stokes problem using a mixed FE formulation

In this contribution, we present an a posteriori error estimator for the incompressible Stokes problem valid for a conventional mixed FE formulation. Due to the saddle-point property of the problem, conventional error estimators developed for pure minimization problems cannot be utilized straight-forwardly. The new estimator is built up by two key ingredients. At first, a computed error approximation, exactly fulfilling the continuity equation for the error, is obtained via local Dirichlet problems. Secondly, we adopt the approach of solving local equilibrated flux-free problems in order to bound the remaining, incompressible, error. In this manner, guaranteed upper and lower bounds, of the velocity “energy norm” of the error as well as goal-oriented (linear) output functionals, with respect to a reference (overkill) mesh are obtained. In particular, it should be noted that this approach requires no computation of hybrid fluxes. Furthermore, the estimator is applicable to mixed FE formulations using continuous pressure approximations, such as the Mini and Taylor–Hood class of elements. In conclusion, a few simple numerical examples are presented, illustrating the accuracy of the error bounds.     

Stability and a posteriori error analysis of discontinuous Galerkin methods for linearized elasticity

We consider discontinuous Galerkin finite element methods for the discretization of linearized elasticity problems in two space dimensions. Inf–sup stability results on the continuous and the discrete level are provided. Furthermore, we derive upper and lower a posteriori error bounds that are robust with respect to nearly incompressible materials, and can easily be implemented within an automatic mesh refinement procedure. The theoretical results are illustrated with a series of numerical experiments.     

Variational formulation of linear problems with nonhomogeneous boundary conditions and internal discontinuities

A variational formulation applicable to linear operators with nonhomogeneous boundary conditions and jump discontinuities is presented. For the formulation to be applicable, the boundary condition and discontinuities have to be consistent with the operator governing the field problem. The problem is set up in a space of suitable continuous bilinear mapping. Thus, operators on inner product spaces, convolution spaces and energy spaces are included as specializations. The basic construction can be used to generate dual-complementary variational principles. Implementation is illustrated by examples. The role of boundary terms in finite element discretizations based on interpolants of limited smoothness is discussed.     

A posteriori error estimation in sensitivity analysis

We present a posteriori error estimators suitable for automatic mesh refinement in the numerical evaluation of sensitivity by means of the finite element method. Both diffusion (Poisson-type) and elasticity problems are considered, and the equivalence between the true error and the proposed error estimator is proved. Application to shape sensitivity is briefly addressed.     

A posteriori error estimators for the fully discrete time dependent Stokes problem with some different boundary conditions

In this paper we study the time dependent Stokes problem with some different boundary conditions. We establish a decoupled variational formulation into a system of velocity and a Poisson equation for the pressure. Hence, the velocity is approximated with curl conforming finite elements in space and Euler scheme in time and the pressure with standard continuous elements in space and Euler scheme in time. Finally, we establish optimal a priori and a posteriori estimates.     

Energy norm a posteriori error estimates for discontinuous Galerkin approximations of the linear elasticity problem

We present a residual-based a posteriori error estimate in an energy norm of the error in a family of discontinuous Galerkin approximations of linear elasticity problems. The theory is developed in two and three spatial dimensions and general nonconvex polygonal domains are allowed. We also present some illustrating numerical examples.     

Integration of second order linear differential equation with mixed boundary conditions

A fourth order finite difference scheme for obtaining an approximate solution of second order linear differential equation lacking the first derivative, with mixed boundary conditions, is presented. The convergence of the method is proved. A numerical illustration is included to demonstrate the practical usefulness of our method.     

Coupling of adaptively refined dual mixed finite elements and boundary elements in linear elasticity

We investigate a coupling of mixed finite elements and Galerkin boundary elements which is stable and leads to symmetric matrices. In the FEM domain, a posteriori error estimates are employed to refine the mesh adaptively. Numerical results are given for plane strain problems.     

A mixed finite element for shell model with free edge boundary conditions Part 1. The mixed variational formulation

After a very brief recall on general shell theory, we construct a mixed variational formulation based on the introduction of a new unknown: the rotation of the normal to the medium surface. In Koiter shell theory (for instance), this rotation can be expressed with respect to the three components of the displacement field of the medium surface and their derivatives. The Lagrange multiplier corresponding to this relation (known as the Kirchhoff-Love kinematical assumption), is also introduced as an independent unknown. There are two main difficulties: one is due to the differential geometry of surfaces and is rather technical; the other is to define correctly the dual space for the Kirchhoff-Love relation. The difficulty is similar to the one met in the characterization of the dual space of the Sobolev space: H¹(ω) (ω being the medium surface of the shell), for which a boundary component appears except for clamped shells which is a very restrictive situation.     

A coupled formulation of finite and boundary element methods in two-dimensional elasticity

A symmetric stiffness formulation based on a boundary element method is studied for the structural analysis of a shear wall, with or without cutouts. To satisfy compatibility requirements with finite beam elements and to avoid problems due to the eventual discontinuities of the traction vector, different interpolation schemes are adopted to approximate the boundary displacements and tractions. A set of boundary integral equations is obtained with the collocation points on the boundary, which are selected by the error minimization technique proposed in this paper, and the stiffness matrix is formulated with those equations and symmetric coupling techniques of finite and boundary element methods. The newly developed plane stress element can have the openings in its interior domain and can be easily linked with the finite beam/column elements.     

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.     

A numerical scheme for mixed boundary value problems in elasticity

A numerical scheme, based upon the Kobayashi-Tranter method with certain modifications, is given for axisymmetric punch and crack problems in elasticity. The problems are reduced to solving a system of linear algebraic equations instead of a Fredholm integral equation of the second kind. A standard program thus allows the treatment of a range of different cases.The indentation of a rigid punch on an elastic layer overlying an elastic foundation is formulated in this fashion and numerical results for various cases are presented.     

A posteriori error estimate for the symmetric coupling of finite elements and boundary elements

In this note we study a posteriori error estimates for a model problem in the symmetric coupling of boundary element and finite elements methods. Emphasis is on the use of the Poincaré-Steklov operator and its discretization which are analyzed in general for both a priori and a posteriori error estimates. Combining arguments from [6] and [9, 10] we refine the a posteriori error estimate obtained in [9, 10]. For quasi-uniform meshes on the boundary, we prove some inequality of a reverse type using techniques from [5] and [36]. This indicates efficiency of the new estimate as illustrated in a numerical example.