1,744 research outputs found
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
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
A class of Petrov-Galerkin finite element methods for the numerical solution of the stationary convection-diffusion equation.
A class of Petrov-Galerkin finite element methods is proposed for the numerical solution of the n dimensional stationary convection-diffusion equation. After an initial review of the literature we describe this class of methods and present both asymptotic and nonasymptotic error analyses. Links are made with the classical Galerkin finite element method and the cell vertex finite volume method. We then present numerical results obtained for a selection of these methods applied to some standard test problems. We also describe extensions of these methods which enable us to solve accurately for derivative values of the solution
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
Proceedings for the ICASE Workshop on Heterogeneous Boundary Conditions
Domain Decomposition is a complex problem with many interesting aspects. The choice of decomposition can be made based on many different criteria, and the choice of interface of internal boundary conditions are numerous. The various regions under study may have different dynamical balances, indicating that different physical processes are dominating the flow in these regions. This conference was called in recognition of the need to more clearly define the nature of these complex problems. This proceedings is a collection of the presentations and the discussion groups
Simulation of stochastic reaction-diffusion processes on unstructured meshes
Stochastic chemical systems with diffusion are modeled with a
reaction-diffusion master equation. On a macroscopic level, the governing
equation is a reaction-diffusion equation for the averages of the chemical
species. On a mesoscopic level, the master equation for a well stirred chemical
system is combined with Brownian motion in space to obtain the
reaction-diffusion master equation. The space is covered by an unstructured
mesh and the diffusion coefficients on the mesoscale are obtained from a finite
element discretization of the Laplace operator on the macroscale. The resulting
method is a flexible hybrid algorithm in that the diffusion can be handled
either on the meso- or on the macroscale level. The accuracy and the efficiency
of the method are illustrated in three numerical examples inspired by molecular
biology
Stability of an upwind Petrov Galerkin discretization of convection diffusion equations
We study a numerical method for convection diffusion equations, in the regime
of small viscosity. It can be described as an exponentially fitted conforming
Petrov-Galerkin method. We identify norms for which we have both continuity and
an inf-sup condition, which are uniform in mesh-width and viscosity, up to a
logarithm, as long as the viscosity is smaller than the mesh-width or the
crosswind diffusion is smaller than the streamline diffusion. The analysis
allows for the formation of a boundary layer.Comment: v1: 18 pages. 2 figures. v2: 22 pages. Numerous details added and
completely rewritten final proof. 8 pages appendix with old proo
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