Next, Tables 5.5 and 5.6 provide the numerical results obtained for Example 3 with (α, ν) = (10, 1) and (α, ν) = (100, 1). As forthe previous examples, e(σ) is again the dominant part of the global error e. Also, we observe that the effectivity indexes remain bounded above and below as the number of degrees of freedom N increases, with bounds close to 1.0, which confirms the reliability of θ and constitutes numerical evidences of its eventual efficiency. In addition, according to the experimental rates of convergence, which are also illustrated by Figures 5.5 and 5.6, the adaptive procedure yields again the quasi-optimal rate of convergence O(h) forthe global error e. Moreover, as expected, the adaptive refinement algorithm is able to identify the singularities of theproblem. In fact, as shown by Meshes 5.5 and 5.6, the adapted meshes are highly refined around the boundary point (1, 1), in whose outer neighborhood the singularity lives. Further, similarly as for Example 2, the refinement is even more localized as α gets larger. Finally, the numerical results concerning Example 4 with (α, ν) = (1000, 0.5) are collected in Table 5.7. The remarks and conclusions here are similar to those for Examples 2 and 3. Again, the effectivity indexes remain bounded, with bounds around 0.11, and, although the experimental rates of convergence of both refinements aproach 1 as N incresases, the global error of the adaptive one begins to decrease before than the uniform one. This fact is clearly observed in Figure 5.7 where the curve e versus N is shown. In addition, as expected, the corresponding adaptive refinement algorithm is able to recognize the inner layer of theproblem. Indeed, as can be seen in Meshes 5.7, the adapted meshes are highly refined around the line x 2 = 0.5 − x 1 . We also notice here that the refinements identify a thin band exactly on this line,
Now, before presenting the examples, we would like to remark in advance that, as compared with more traditional mixed methods, and besides the fact, already emphasized, of being able to choose any ﬁnite element subspace, our augmented approach presents other important advantages, as well. Indeed, let us ﬁrst observe that in the case of uniform reﬁnements each interior edge (resp. interior node) belongs to 2 (resp. 6) triangles, which yields corresponding correction factors of 1 2 and 1 6 when counting the global number of degrees of freedom, say N , in terms of the number of triangles, say M . Then, it is not diﬃcult to see that the number of unknowns N of (10) behaves asymptotically as 5 M , whereas this behaviour is given by 7 . 5 M when the well-known PEERS from  is used in the Galerkin scheme of the non-augmented formulation. In other words, the discrete system using PEERS introduces about 50% more degrees of freedom than our approach at each mesh, and therefore the augmented method becomes a much cheaper alternative. Furthermore, it is important to note that the polynomial degrees involved in the deﬁnition of H h σ × H u
We extend the applicability of the augmented dual-mixedmethod introduced recently in [4, 5] to theproblem of linear elasticity with mixed boundary conditions. Themethod is based on the Hellinger–Reissner principle and the symmetry of the stress tensor is imposed in a weak sense. The Neuman boundary condition is prescribed in thefiniteelement space. Then, suitable Galerkin least-squares type terms are added in order to obtain an augmented variational formulation which is coercive in the whole space. This allows to use any finiteelement subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.
In 1979 the Winter Annual Meeting of the ASME was held in New York. In this meeting early upwind finiteelement formulations were reviewed  and other new techniques were proposed. Belytschko and Eldib  introduced an amplification scheme for achieving an upwind finiteelement formulation. However, the behavior of the amplification scheme was shown to be similar to other upwind finiteelement methods. In this meeting, the idea of introducing numerical dissipation only along the streamlines is pointed out by Griffiths and Mitchell  and formally stated by Hughes and Brooks . This is the main idea underlying the streamline upwind Petrov-Galerkin method . Almost simultaneously Kelly et al.  suggested the same procedure to eliminate crosswind diffusion in multidimensional solutions. This method can be formulated by using a modified weighting function forthe convective term only. This scheme was called streamline upwind method.
Despite DIR is widely used in the medical imaging community, the mathematical and numerical analysis of DIR remains understudied. The DIR continuous problem has been formulated using mainly three approaches: minimization of similarity measures (with or without constraints), as an optimal mass transport problem [22, 31, 11], or as a level set segmentation-registration combined problem [42, 17]. Theproblem of minimizing similarity measures has been studied in [5, 43], where the direct method of calculus of variations has been used to establish existence of solutions. The optical flow formulation, an associated problem which can be seen as a sequence of registration problems in time, was proposed by Horn & Schunk in 1980 , and has been the subject of analysis from an optimal- control problem point of view [7, 27]. Well-posedness of optical flow schemes has been established for Dirichlet boundary conditions under reasonable assumptions [18, 41]. Besides providing existence and uniqueness of the solution, by assuming only uniform boundedness on the images, these studies show that the solution is a step-wise diffeomorphism, which is a desirable regularity property when it comes to warping images. The analysis of the numerical schemes proposed to solve similarity- minimization formulations has received less attention. A noteworthy approach is the work of P¨ oschl et al. , where both the continuous and discretized problems are analyzed, and a solution is found using a primal finite-element approximation that is shown to be convergent. However, the analysis is restricted to polyconvex energy densities (both forthe similarity measure and regularizer) and volume-preserving transformations, and does not account forthe convergence of the transformation gradients and stresses. A more traditional Galerkin approach has been introduced in  for optimal- control-based registration, but requires a considerable degree of regularity (H 2+δ ) of the target and reference image functions, not required by other traditional formulations. While most approaches to DIR problems are based on primal formulations, a mixed formulation of the similarity minimization problem has been proposed in the setting of fluid registration schemes [12, 35], where a sequence of incompressible Stokes problems are solved to find the optimal displacement and pressure fields. While directly solving forthe pressure field, which is desirable to understand the mechanical behavior of the images being registered, limited analysis has been provided to understand the well-posedness of the continuous problem and convergence of numerical discretizations of mixed formulations of DIR problems that use elastic regularizers.
4. Convergence of the MLG–BDF2 method. In this section we perform the error analysis of themethod following a step by step approach. First, we re- call auxiliary results concerning the convergence of the semidiscrete Stokesproblem; second, we study the error of the approximation of the departure points and some related results, considering that the system (3.4) is integrated by a Runge–Kutta scheme of order r ≥ 2; third, we end up establishing the convergence of themethod in the l ∞ (L 2 (D)) and l ∞ (H 1 (D)) norms forthe velocity and l 2 (L 2 (D)) norm forthe pressure. In the developments that follow we need the ﬁnite dimensional space V h
Deformable Image Registration (DIR) is a powerful computational methodfor image analysis, with promising applications in the diagnosis of human disease. Despite being widely used in the medical imaging community, the mathematical and numerical analysis of DIR methods still has many open questions. Further, recent applications of DIR in- clude the quantification of mechanical quantities in addition to the aligning transformation, which justifies the development of novel DIR formulations for which the accuracy and convergence of fields other than the aligning transformation can be studied. In this work we propose and analyze a primal, mixed and augmented formulations forthe DIR prob- lem, together with their finite-element discretization schemes for their numerical solution. The DIR variational problem is equivalent to the linear elasticity problem with a source term that has a nonlinear dependence on the unknown field. Fixed point arguments and small data assumptions are employed to derive the well-posedness of both the continuous and discrete schemes forthe usual primal and mixed variational formulations, as well as for an augmented version of the later. In particular, continuous piecewise linear elements forthe displacement in the case of the primal method, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) forthe stress together with piecewise constants (resp. continuous piecewise linear) forthe displacement when using themixed approach (resp. its augmented version), constitute feasible choices that guarantee the stability of the asso- ciated Galerkin systems. A priori error estimates derived by using Strang-type lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of themethod.
The following paper shows a FiniteElement formulation forthe resolution of the – local and convective acceleration terms including- Navier-Stokes equations, which gives analytical response to theproblem of viscous, incompressible, unsteady flows. The integration of the resulting non-linear system of first order ordinary differential equations, is made upon a successive approximation algorithm together with an implicit backward time integrating scheme. The interpolation of the spatial domain is made in terms of a Q1/P0 pair (bilinear velocity-constant pressure). The usage of a Bubnov Galerkin formulation in the process of obtaining a weak form implies that flows of a certain velocity need the employment of a very refined spatial mesh so as to avoid numerical instability. For high Reynolds numbers the convection term becomes predominant compared to the diffussion term and a different algorithm (SPGU, GLS), should be introduced. Finally the developed program is checked over some of the most commonly used flow tests and its results on velocity and pressure are shown.
Theproblem of Darcy flow is of great importance in civil, geotechnical and petroleum engineering. It describes the flow of a fluid through a porous medium. The natural unknowns are the fluid pressure and the fluid velocity, being the latter the unknown of primary interest in many applications. Theproblem can be reduced to an elliptic equation forthe pressure with a Neumann boundary condition. Although this reduced problem can be solved with appropriate accuracy by a classical Galerkin finiteelementmethod, typically there is a loss of accuracy in the approximation of the velocity through the pressure gradients. Moreover, with this reduced formulation, local mass conservation is not guaranteed. For this reason, the primal formulation forthe pressure is not considered adequate for practical engineering applications.
On the other hand, we recall that the application of adaptive algorithms, based on a posteriori error estimates, usually guarantees the quasi-optimal rate of convergence of the ﬁnite element solution to boundary value problems. In addition, this adaptivity is specially necessary for nonlinear problems where no a priori hints on how to build suitable meshes are available. To this respect, we have shown recently that the combination of the usual Bank–Weiser approach from  with the analysis from [3,4] allows to derive fully explicit and reliable a posteriori error estimates forthe dual-mixed variational formulations (showing a two- fold saddle point structure) of some linear and nonlinear problems (see, e.g. [2,6,7]). However, no a pos- teriori error analysis has been developed yet forthe nonlinear Stokes problems studied in . Therefore, as a natural continuation of our results in , in the present paper we apply the Bank–Weiser type a posteriori error analysis mentioned above to derive reliable estimates forthemixed ﬁnite element scheme (1.6). The rest of this work is organized as follows. In Section 2 we collect some basic results on Sobolev spaces and state the main result of this paper. The proof of our a posteriori estimate, which makes use of the Ritz projection of the error, is provided in Section 3. In Section 4 we prove the quasi-eﬃciency of the estimator and discuss on suitable choices forthe auxiliary functions needed for its computation. Finally, several numerical results illustrating the good performance of the adaptive algorithm are reported in Section 5.
In this paper we adapt to the vibration problemthemixedfiniteelementmethod proposed and analyzed by Arunakirinathar and Reddy in  forthe load problemfor elastic curved rods. With this purpose, we settle the corresponding spectral problem by including the mass terms arising from displacement and rotational inertia in the model, as proposed in . Our assumptions on the rods are slightly weaker than those in . On the one hand we allow for non-constant geometric and physical coefficients varying smoothly along the rod. On the other hand, we do not assume that the Frenet basis associated with the line of cross-section centroids is a set of principal axes. We prove that the resulting method yield optimal order approximation of displacements and rotations of the vibration modes, as well as a double order of convergence forthe vibration frequencies. Under mild assumptions, we also prove that the error estimates do not degenerate as the thickness becomes small, which allow us to conclude that themethod is locking free.
Themixed ﬁnite elementmethod proposed in the present paper simply relies on the introduction of the stress and gradient of the velocity tensors as auxiliary unknowns, and it does not require any inversion process, whence the resulting variational formulation shows, as in [3,16], a twofold saddle point structure. Therefore, the abstract theory for this kind of operator equation (see, e.g. [8,9,11,14,15]), which constitutes a generalization of the well known Babu ska–Brezzi theory, can also be applied to the present situation. In particular, eﬃcient iterative methods to solve the associated linear systems are available (see, e.g. [12,13]). The extension of this approach to kinematic viscosity functions not satisfying (1.2) or (1.3), which includes the Carreau law with j 0 ¼ 0 or b > 2, will be reported in a separate work.
An application of theFiniteElementMethod (FEM) to the solution of a geometric problem is shown. Theproblem is related to curve fitting i.e. pass a curve trough a set of given points even if they are irregularly spaced. Situations where cur ves with cusps can be encountered in the practice and therefore smooth interpolatting curves may be unsuitable. In this paper the possibilities of the FEM to deal with this type of problems are shown. A particular example of application to road planning is discussed. In this case the funcional to be minimized should express the unpleasent effects of the road traveller. Some comparative numerical examples are also given.
Forthe introduced example, using a force on the structure surface this approach has been tested. The uncoupled solution was taken as a starting point, and four solution update steps were per- formed. After the last solution update, the BiCG solver was used to finally calculate the coupled solution, and the numerical error of each intermediate solution (relative to the coupled BiCG so- lution) was analyzed.
The plate is supposed to be steel ASTM A36, whose physical Young´s modulus is es E = 2x1011 Pa. According to the configuration presented in the Fig. 1, It is intended to estimate the actual tensions distribution in a stress concentrator, along the Y axis, carrying out the following procedure:
The main result of this paper concerns contributions to the analysis of a fully dis- crete nonlinear FEM–BEM formulation.We had to answer two principal diﬃculties. On the one hand, unless the nonlinear operator is Lipschitz continuous and strongly monotone, Strang’s lemma is not satisﬁed and there is no general framework to control
An Uzawa-type algorithm forthe saddle point system, coupled with a domain decomposition approach, has been proposed in [ 12 ] . The original system is transformed into a new system by augmentation with the scalar Laplacian as a weight matrix, and it is shown that the condition number of the resulting preconditioned system grows logarithmically with respect to the ratio between the subdomain diameter and the mesh size. Themethod incorporates augmentation and is parameter dependent. Its convergence properties rely on extreme eigenvalues of the augmented Schur complement, which may be difficult to evaluate.
Besides the propagating problem of electromagnetic waves in matter, this work mainly concerns with the effects undergone when an EMW goes throw a discontinuity in the propagating media. It is well known that when an EMW encounters a interface between two media, a fraction of the energy associate to the incident wave is reflected from the interface, and the rest of the energy is transmitted throw the interface, given place to a reflected and transmitted waves, respectively. In solving theproblem of the relative amplitudes of the reflected and transmitted waves to the incident wave, we are concerned with a boundary conditions problem. The relation of the reflected and transmitted amplitudes to the incident amplitude is governed by the Maxwell’s equations.
In the rest of the paper we consider a coupled system consisting of an acoustic fluid (i.e. inviscid compressible barotropic) in contact with an elastic porous medium. Both are enclosed in a three-dimensional cavity with rigid walls except one on which a harmonic excitation is applied. Let Ω F and Ω A be the domains occupied by the fluid and the porous medium, respectively (see Figure 1).