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Displaying 61 –
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105
We investigate unilateral contact problems with cohesive forces, leading to
the constrained minimization of a possibly nonconvex functional. We
analyze the mathematical structure of the minimization problem.
The problem is reformulated in terms of a three-field augmented
Lagrangian, and sufficient conditions for the existence of a local
saddle-point are derived. Then, we derive and analyze mixed finite
element approximations to the stationarity conditions of the three-field
augmented Lagrangian....
The discretisation of the Oseen problem by finite element methods may suffer
in general from two shortcomings. First, the discrete inf-sup
(Babuška-Brezzi)
condition can be violated. Second, spurious oscillations
occur due to the dominating convection. One way to overcome both
difficulties is the use of local projection techniques. Studying
the local projection method in an abstract setting, we show that
the fulfilment of a local inf-sup condition between approximation and
projection spaces...
In this paper we propose a solution of the Lambertian shape-from-shading
(SFS) problem by designing
a new mathematical framework based on the
notion of viscosity solution. The power of our approach is twofolds:
(1) it defines a notion of weak solutions
(in the viscosity sense) which does not
necessarily require boundary data. Moreover, it allows to characterize the
viscosity solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, SIAM J. Numer. Anal.29 (1992) 867–884],...
This paper provides an accelerated two-grid stabilized mixed finite element scheme for the Stokes eigenvalue problem based on the pressure projection. With the scheme, the solution of the Stokes eigenvalue problem on a fine grid is reduced to the solution of the Stokes eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. By solving a slightly different linear problem on the fine grid, the new algorithm significantly improves the theoretical error...
We consider a general abstract framework of a continuous elliptic
problem set on a Hilbert space V that is approximated by a family of (discrete) problems
set on a finite-dimensional space of finite dimension not
necessarily included into V. We give a series of realistic
conditions on an error estimator that allows to conclude that the
marking strategy of bulk type leads to the geometric convergence
of the adaptive algorithm. These conditions are then verified for
different concrete problems...
In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...
In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...
We generalize the overlapping Schwarz domain decomposition method to problems of linear elasticity. The convergence rate independent of the mesh size, coarse-space size, Korn’s constant and essential boundary conditions is proved here. Abstract convergence bounds developed here can be used for an analysis of the method applied to singular perturbations of other elliptic problems.
An alternating-direction iterative procedure is described for a class of Helmholz-like problems. An algorithm for the selection of the iteration parameters is derived; the parameters are complex with some having positive real part and some negative, reflecting the noncoercivity and nonsymmetry of the finite element or finite difference matrix. Examples are presented, with an applications to wave propagation.
The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error...
The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is
𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error...
We present the current Reduced Basis framework for the efficient numerical approximation of parametrized steady Navier–Stokes equations. We have extended the existing setting developed in the last decade (see e.g. [S. Deparis, SIAM J. Numer. Anal. 46 (2008) 2039–2067; A. Quarteroni and G. Rozza, Numer. Methods Partial Differ. Equ. 23 (2007) 923–948; K. Veroy and A.T. Patera, Int. J. Numer. Methods Fluids 47 (2005) 773–788]) to more general affine and nonaffine parametrizations (such as volume-based...
In this paper, we discuss an hp-discontinuous Galerkin finite
element method (hp-DGFEM) for the laser surface hardening of
steel, which is a constrained optimal control problem governed by a
system of differential equations, consisting of an ordinary
differential equation for austenite formation and a semi-linear
parabolic differential equation for temperature evolution. The space
discretization of the state variable is done using an hp-DGFEM,
time and control discretizations are based on a discontinuous
Galerkin...
In this paper, we discuss an hp-discontinuous Galerkin finite
element method (hp-DGFEM) for the laser surface hardening of
steel, which is a constrained optimal control problem governed by a
system of differential equations, consisting of an ordinary
differential equation for austenite formation and a semi-linear
parabolic differential equation for temperature evolution. The space
discretization of the state variable is done using an hp-DGFEM,
time and control discretizations are based on a discontinuous
Galerkin...
We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of...
We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution.
The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L2-stability of the discrete advection operator provided it...
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