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A continuous finite element method to approximate Friedrichs' systems is
proposed and analyzed. Stability is achieved by penalizing the jumps
across mesh
interfaces of the normal derivative of some components of the discrete solution.
The convergence analysis leads to optimal convergence rates
in the graph norm and suboptimal of order ½ convergence rates in
the L2-norm. A variant of the method specialized to
Friedrichs' systems associated with elliptic PDE's in mixed form and
reducing the number...
In this article we introduce an adaptive multi-level
method in space and time for convection diffusion problems. The scheme
is based on a multi-level spatial splitting and the use of different
time-steps. The temporal discretization relies on the characteristics method.
We derive an a posteriori error estimate and design a corresponding
adaptive algorithm.
The efficiency of the multi-level method is illustrated by numerical experiments,
in particular for a convection-dominated problem.
We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly...
We consider the double-diffusive convection phenomenon and analyze the governing equations. A system of partial differential equations describing the convective flow arising when a layer of fluid with a dissolved solute is heated from below is considered. The problem is placed in a functional analytic setting in order to prove a theorem on existence, uniqueness and continuous dependence on initial data of weak solutions in the class . This theorem enables us to show that the infinite-dimensional...
In this paper, we study the boundary penalty method for optimal control of unsteady Navier–Stokes type system that has been proposed as an alternative for Dirichlet boundary control. Existence and uniqueness of solutions are demonstrated and existence of optimal control for a class of optimal control problems is established. The asymptotic behavior of solution, with respect to the penalty parameter ϵ, is studied. In particular, we prove convergence of solutions of penalized control problem to the...
The subject of the paper is the derivation of error estimates for the combined finite volume-finite element method used for the numerical solution of nonstationary nonlinear convection-diffusion problems. Here we analyze the combination of barycentric finite volumes associated with sides of triangulation with the piecewise linear nonconforming Crouzeix-Raviart finite elements. Under some assumptions on the regularity of the exact solution, the and error estimates are established. At the end...
We study a family of non linear schemes for the numerical solution of linear advection on arbitrary grids in several space dimension. A proof of weak convergence of the family of schemes is given, based on a new Longitudinal Variation Diminishing (LVD) estimate. This estimate is a multidimensional equivalent to the well-known TVD estimate in one dimension. The proof uses a corollary of the Perron-Frobenius theorem applied to a generalized Harten formalism.
We study a family of non linear schemes for the numerical solution of
linear advection on arbitrary grids in several space dimension.
A proof of weak convergence of the family of schemes is given, based on a new Longitudinal Variation Diminishing (LVD) estimate.
This estimate is a multidimensional equivalent to the well-known TVD estimate in one dimension. The proof uses a corollary of the Perron-Frobenius theorem applied to a generalized Harten formalism.
The model order reduction methodology of reduced basis (RB)
techniques offers efficient treatment of parametrized partial differential
equations (P2DEs) by providing both approximate solution procedures and
efficient error estimates.
RB-methods have so far mainly been applied to finite element schemes
for elliptic and parabolic problems. In the current study
we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations....
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