712 research outputs found
Monotone difference schemes for weakly coupled elliptic and parabolic systems
The present paper is devoted to the development of the theory of monotone difference schemes, approximating the so-called weakly coupled system of linear elliptic and quasilinear parabolic equations. Similarly to the scalar case, the canonical form of the vector-difference schemes is introduced and the definition of its monotonicity is given. This definition is closely associated with the property of non-negativity of the solution. Under the fulfillment of the positivity condition of the coefficients, two-side estimates of the approximate solution of these vector-difference equations are established and the important a priori estimate in the uniform norm C is given
МОНОТОННЫЕ РАЗНОСТНЫЕ СХЕМЫ ДЛЯ СИСТЕМ ЭЛЛИПТИЧЕСКИХ И ПАРАБОЛИЧЕСКИХ УРАВНЕНИЙ
In this article, for the canonical form of vector-difference schemes under the positivity conditions of matrix coefficients the two-sided estimates for an approximate solution at the arbitrary non sign- constant input data of the problem are obtained. The obtained results are used for deriving two-swided estimates and a priori estimates in the norm C of monotone vector-difference schemes that approximate the weakly coupled systems of elliptic and parabolic equations with the Dirichlet foundary conditions.В настоящей работе для канонической формы векторно-разностных схем общего вида при условиях положительности матричных коэффициентов получены двусторонние оценки сеточного решения при произвольных незнакопостоянных входных данных задачи. Полученные результаты применяются для получения двусторонних оценок и априорных оценок в норме С конкретных монотонных векторно-разностных схем, аппроксимирующих слабо связанные системы эллиптических и параболических уравнений с граничными условиями Дирихле
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
Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements
The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical
Methods for Singularly Perturbed Differential Equations" appeared many years
ago and was for many years a reliable guide into the world of numerical methods
for singularly perturbed problems. Since then many new results came into the
game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827
Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions
We analytically and numerically analyze groundwater flow in a homogeneous
soil described by the Richards equation, coupled to surface water represented
by a set of ordinary differential equations (ODE's) on parts of the domain
boundary, and with nonlinear outflow conditions of Signorini's type. The
coupling of the partial differential equation (PDE) and the ODE's is given by
nonlinear Robin boundary conditions. This article provides two major new
contributions regarding these infiltration conditions. First, an existence
result for the continuous coupled problem is established with the help of a
regularization technique. Second, we analyze and validate a solver-friendly
discretization of the coupled problem based on an implicit-explicit time
discretization and on finite elements in space. The discretized PDE leads to
convex spatial minimization problems which can be solved efficiently by
monotone multigrid. Numerical experiments are provided using the DUNE numerics
framework.Comment: 34 pages, 5 figure
- …