Volume 16, Issue 1

Classical solutions of mixed problems for first order partial functional differential systems in several independent variables are approximated in the paper with solutions of a difference problem of the Euler type. The mesh for the approximate solutions is obtained by a numerical solving of equations of characteristics. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from a general model by specializing given operators.

For a general diffusion-convection-reaction equation, we analyze families of nonconforming finite elements of arbitrary order on a sequence of multilevel grids consisting of quadrilaterals or hexahedra. We prove existence and uniqueness of the discrete solution and optimal order of convergence in the broken H 1 -seminorm and the L 2 -norm. The novelty of our approach is that a new integral compatibility condition of the discrete functions across the element faces is introduced such that it can be solely treated on the reference element once for all faces of the grid. A numerical comparison between conforming and nonconforming discretizations will be given in the three-dimensional case.

We consider variational inequalities related to problems with nonlinear boundary conditions. We are focused on deriving a posteriori estimates of the difference between exact solutions of such type variational inequalities and any function lying in the admissible functional class of the problem considered. These estimates are obtained by an advanced version of the variational approach earlier used for problems with uniformly convex functionals (see [28,30]). It is shown that the structure of error majorants reflects properties of the exact solution. The majorants provide guaranteed upper bounds of the error for any conforming approximation and possess necessary continuity properties. In the series of numerical tests performed, it was shown that the estimates are explicitly computable, provide sharp bounds of approximation errors, and give high quality indication of the distribution of local (elementwise) errors.