A Convergence Analysis of Hopscotch Methods for Fourth Order Parabolic Equations.
A discrete maximum principle [Book]
A note on nonhomogeneous initial and boundary conditions in parabolic problems solved by the Rothe method
When solving parabolic problems by the so-called Rothe method (see K. Rektorys, Czech. Math. J. 21 (96), 1971, 318-330 and other authors), some difficulties of theoretical nature are encountered in the case of nonhomogeneous initial and boundary conditions. As a rule, these difficulties lead to rather unnatural additional conditions imposed on the corresponding bilinear form and the initial and boundary functions. In the present paper, it is shown how to remove such additional assumptions in the...
An enclosure generating modification of the method of discretization in time
An extension of Rothe's method to non-cylindrical domains
In this paper Rothe’s classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied.
Computational studies of conserved mean-curvature flow
The paper presents the results of numerical solution of the evolution law for the constrained mean-curvature flow. This law originates in the theory of phase transitions for crystalline materials and describes the evolution of closed embedded curves with constant enclosed area. It is reformulated by means of the direct method into the system of degenerate parabolic partial differential equations for the curve parametrization. This system is solved numerically and several computational studies are...
Computational studies of non-local anisotropic Allen-Cahn equation
The paper presents the results of numerical solution of the Allen-Cahn equation with a non-local term. This equation originally mentioned by Rubinstein and Sternberg in 1992 is related to the mean-curvature flow with the constraint of constant volume enclosed by the evolving curve. We study this motion approximately by the mentioned PDE, generalize the problem by including anisotropy and discuss the computational results obtained.
Continuous approximation of time-periodic solutions of a linear parabolic equation
Contractivity of Locally One-Dimensional Splitting Methods.
Energy error estimates for a linear scheme to approximate nonlinear parabolic problems
Forced anisotropic mean curvature flow of graphs in relative geometry
The paper is concerned with the graph formulation of forced anisotropic mean curvature flow in the context of the heteroepitaxial growth of quantum dots. The problem is generalized by including anisotropy by means of Finsler metrics. A semi-discrete numerical scheme based on the method of lines is presented. Computational results with various anisotropy settings are shown and discussed.
Free Boundary Problems with Nonlinear Source Terms.
Interface dynamics for an anisotropic Allen-Cahn equation
La méthode des caractéristiques pour les problèmes de convection-diffusion stationnaires
Numerical and theoretical treating of evolution problems by the method of discretization in time
On existence of the weak solution for nonlinear diffusion equation
The paper concerns the existence of bounded weak solutions of a anonlinear diffusion equation with nonhomogeneous mixed boundary conditions.
On -convergence of Rothe’s method
On the application of mixed finite element methods to the wave equations
Optimal grids for anisotropic problems.
Second-Order Finite Element Approximations and a posteriori Error Estimation for Two-Dimensional Parabolic Systems.