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In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete...
In this work we introduce a new class of lowest order methods for
diffusive problems on general meshes with only one unknown per
element.
The underlying idea is to construct an incomplete piecewise affine
polynomial space with optimal approximation properties starting
from values at cell centers.
To do so we borrow ideas from multi-point finite volume methods,
although we use them in a rather different context.
The incomplete polynomial space replaces classical complete
polynomial spaces...
We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of...
We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of...
We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of...
The paper is devoted to the convergence analysis of a well-known
cell-centered Finite Volume Method (FVM) for a
convection-diffusion problem in . This FVM is based on Voronoi
boxes and
exponential fitting. To prove the convergence of the FVM, we use
a new nonconforming Petrov-Galerkin Finite Element Method (FEM)
for which the system of linear equations coincides completely with
that of the FVM. Thus, by proving convergence properties of the
FEM we obtain similar ones for the FVM. For the error...
We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows.
Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes.
Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup...
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...
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