The fundamental principles of adaptive grid generation for the numerical analysis of physical phenomena described by systems of partial differential equations are examined in an analytical review. Topics addressed include weight functions, equidistribution in one dimension, the specification of coefficients in the linear weight, the attraction to a given grid on a curve, evolutionary forces, and metric notation. Consideration is given to curve-by-curve methods, finite-volume methods, variational methods, and temporal aspects.

Adaptive grid generation

We discuss a moving-mesh finite element method for solving initial boundary value problems for vector systems of partial differential equations in one space dimension and time. The system is discretized using piecewise linear finite element approximations in space and a backward difference code for stiff ordinary differential systems in time. A spatial-error estimation is calculated using piecewise quadratic approximations that use the superconvergence properties of parabolic systems to gain computational efficiency. The spatial-error estimate is used to move and locally refine the finite element mesh in order to equidistribute a measure of the total spatial error and to satisfy a prescribed error tolerance. Ordinary differential equations for the spatial-error estimate and the mesh motion are integrated in time using the same backward difference software that is used to determine the numerical solution of the partial differential system. We present several details of an algorithm that may be used to develop a general-purpose finite element code for one-dimensional parabolic partial differential systems. The algorithm combines mesh motion and local refinement in a relatively efficient manner and attempts to eliminate problemdependent numerical parameters. A variety of examples that motivate our mesh-moving strategy and illustrate the performance of our algorithm are presented.

A moving-mesh finite element method with local refinement for parabolic partial differential equations

Algebraic dynamic multilevel (ADM) method for fully implicit simulations of multiphase flow in porous media

Summary. In the present review the authors consider the application of mathematical modeling methods for the study of laser radiation influence on absorbing materials. The principle attention is paid to the mathematical formulations o f the phenomenon studied as well as the analysis o f the simulation results. The numerical solution methods are analysed briefly. It is shown that the mathematical modeling is the main method for theoretical studies of dynamically non-equilibrium phase transformations and processes o f plasma formation, which are the basis of numerous kinds o f laser treatment of materials.

http://samarskii.ru/articles/1994/1994-003ocr.pdf

Mathematical modeling in the technology of laser treatments of materials

A two-dimensional mesh moving technique for time-dependent partial differential equations

An Adaptive Grid Finite Difference Method for Conservation Laws

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An Adaptive Grid Finite Difference Method for Conservation Laws