730 research outputs found
An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems
Heterogeneous anisotropic diffusion problems arise in the various areas of
science and engineering including plasma physics, petroleum engineering, and
image processing. Standard numerical methods can produce spurious oscillations
when they are used to solve those problems. A common approach to avoid this
difficulty is to design a proper numerical scheme and/or a proper mesh so that
the numerical solution validates the discrete counterpart (DMP) of the maximum
principle satisfied by the continuous solution. A well known mesh condition for
the DMP satisfaction by the linear finite element solution of isotropic
diffusion problems is the non-obtuse angle condition that requires the dihedral
angles of mesh elements to be non-obtuse. In this paper, a generalization of
the condition, the so-called anisotropic non-obtuse angle condition, is
developed for the finite element solution of heterogeneous anisotropic
diffusion problems. The new condition is essentially the same as the existing
one except that the dihedral angles are now measured in a metric depending on
the diffusion matrix of the underlying problem. Several variants of the new
condition are obtained. Based on one of them, two metric tensors for use in
anisotropic mesh generation are developed to account for DMP satisfaction and
the combination of DMP satisfaction and mesh adaptivity. Numerical examples are
given to demonstrate the features of the linear finite element method for
anisotropic meshes generated with the metric tensors.Comment: 34 page
Finite volume schemes for diffusion equations: introduction to and review of modern methods
We present Finite Volume methods for diffusion equations on generic meshes,
that received important coverage in the last decade or so. After introducing
the main ideas and construction principles of the methods, we review some
literature results, focusing on two important properties of schemes (discrete
versions of well-known properties of the continuous equation): coercivity and
minimum-maximum principles. Coercivity ensures the stability of the method as
well as its convergence under assumptions compatible with real-world
applications, whereas minimum-maximum principles are crucial in case of strong
anisotropy to obtain physically meaningful approximate solutions
Piecewise linear transformation in diffusive flux discretization
To ensure the discrete maximum principle or solution positivity in finite
volume schemes, diffusive flux is sometimes discretized as a conical
combination of finite differences. Such a combination may be impossible to
construct along material discontinuities using only cell concentration values.
This is often resolved by introducing auxiliary node, edge, or face
concentration values that are explicitly interpolated from the surrounding cell
concentrations. We propose to discretize the diffusive flux after applying a
local piecewise linear coordinate transformation that effectively removes the
discontinuities. The resulting scheme does not need any auxiliary
concentrations and is therefore remarkably simpler, while being second-order
accurate under the assumption that the structure of the domain is locally
layered.Comment: 11 pages, 1 figures, preprint submitted to Journal of Computational
Physic
Semi-Lagrangian methods for parabolic problems in divergence form
Semi-Lagrangian methods have traditionally been developed in the framework of
hyperbolic equations, but several extensions of the Semi-Lagrangian approach to
diffusion and advection--diffusion problems have been proposed recently. These
extensions are mostly based on probabilistic arguments and share the common
feature of treating second-order operators in trace form, which makes them
unsuitable for mass conservative models like the classical formulations of
turbulent diffusion employed in computational fluid dynamics. We propose here
some basic ideas for treating second-order operators in divergence form. A
general framework for constructing consistent schemes in one space dimension is
presented, and a specific case of nonconservative discretization is discussed
in detail and analysed. Finally, an extension to (possibly nonlinear) problems
in an arbitrary number of dimensions is proposed. Although the resulting
discretization approach is only of first order in time, numerical results in a
number of test cases highlight the advantages of these methods for applications
to computational fluid dynamics and their superiority over to more standard low
order time discretization approaches
Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure
We present a numerical method for approximating the solutions of degenerate
parabolic equations with a formal gradient flow structure. The numerical method
we propose preserves at the discrete level the formal gradient flow structure,
allowing the use of some nonlinear test functions in the analysis. The
existence of a solution to and the convergence of the scheme are proved under
very general assumptions on the continuous problem (nonlinearities, anisotropy,
heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the
efficiency and of the robustness of our approach
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