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In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by a simple post processing. Based on the auxiliary solution, we define the adaptive algorithm which guides...
We prove convergence and quasi-optimal complexity of an adaptive finite element algorithm on triangular meshes with standard mesh refinement. Our algorithm is based on an adaptive marking strategy. In each iteration, a simple edge estimator is compared to an oscillation term and the marking of cells for refinement is done according to the dominant contribution only.
In addition, we introduce an adaptive stopping criterion for iterative solution which compares an estimator for the iteration error...
Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical...
Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered.
The non-smoothness arises from a L1-norm in the objective functional.
The problem is regularized to permit the use of the semi-smooth Newton
method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical...
We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions...
The standard discretization of the Stokes and
Navier–Stokes equations in vorticity and stream function formulation by affine finite
elements is known for its bad convergence. We present here a modified discretization, we
prove that the convergence is improved and we establish a priori error estimates.
In this paper, the convergence of a
Neumann-Dirichlet algorithm to approximate Coulomb's contact
problem between two elastic bodies is proved in a continuous setting. In this algorithm, the natural interface between the two bodies is retained as a decomposition zone.
An equilibrium triangular block-element, proposed by Watwood and Hartz, is subjected to an analysis and its approximability property is proved. If the solution is regular enough, a quasi-optimal error estimate follows for the dual approximation to the mixed boundary value problem of elasticity (based on Castigliano's principle). The convergence is proved even in a general case, when the solution is not regular.
A semi-coercive problem with unilateral boundary conditions of the Signoriti type in a convex polygonal domain is solved on the basis of a dual variational approach. Whereas some strong regularity of the solution has been assumed in the previous author’s results on error estimates, no assumption of this kind is imposed here and still the -convergence is proved.
The numerical solution of the Hartree-Fock equations is a central problem in quantum chemistry for which numerous algorithms exist. Attempts to justify these algorithms mathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod. Numer. Anal. 34 (2000) 749–774], but, to our knowledge, no complete convergence proof has been published, except for the large-Z result of [M. Griesemer and F. Hantsch, Arch. Rational Mech. Anal. (2011) 170]. In this paper, we prove the convergence of...
The numerical solution of the Hartree-Fock equations is a central problem in quantum chemistry for which numerous algorithms exist. Attempts to justify these algorithms mathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod. Numer. Anal. 34 (2000) 749–774], but, to our knowledge, no complete convergence proof has been published, except for the large-Z result of [M. Griesemer and F. Hantsch, Arch. Rational Mech. Anal. (2011) 170]. In this paper, we prove the convergence of...
The numerical solution of the Hartree-Fock equations is a central problem in quantum
chemistry for which numerous algorithms exist. Attempts to justify these algorithms
mathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod.
Numer. Anal. 34 (2000) 749–774], but, to our knowledge, no
complete convergence proof has been published, except for the large-Z
result of [M. Griesemer and F. Hantsch, Arch. Rational Mech. Anal. (2011)
...
This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported.
We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear)
interface problem for the 2D Laplacian. We introduce some new a posteriori
error estimators based on the (h − h/2)-error
estimation strategy. In particular, these include the approximation error for the boundary
data, which allows to work with discrete boundary integral operators only. Using the
concept of estimator reduction, we prove that the proposed adaptive...
We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear)
interface problem for the 2D Laplacian. We introduce some new a posteriori
error estimators based on the (h − h/2)-error
estimation strategy. In particular, these include the approximation error for the boundary
data, which allows to work with discrete boundary integral operators only. Using the
concept of estimator reduction, we prove that the proposed adaptive...
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