Method of lines and conservation of nonnegativity

Generally speaking, a parabolic problem conserves nonnegativity if nonnegative input data lead to a nonnegative solution. This property of the mathematical model is important in physics if we deal with absolute temperature, concentration, density etc. The well-known comparison principle guarantees that the homogeneous linear parabolic problem with homogeneous Dirichlet boundary condition has nonnegative solution for any nonnegative initial condition. It is shown that the standard semidiscretization of this problem, namely the method of lines combined with the first order finite element method, does not conserve nonnegativity

Complementary error bounds for elliptic systems and applications.

This contribution derives guaranteed upper bounds of the energy norm of the approximation error for linear elliptic partial differential systems. We generalize the complementarity error estimates known for scalar elliptic problems to general diffusion-convection-reaction linear elliptic systems. For systems we prove analogous properties of these error bounds as for the scalar case. A brief description how the presented general theory applies to linear elasticity is included as well as an application to chemical systems with reactions of at most first order. Numerical experiments showing the sharpness of the obtained upper bounds and their behavior in the adaptive procedure are presented, too. © 2012 Elsevier Inc. All rights reserved

Complementary error bounds for elliptic systems and applications.

This contribution derives guaranteed upper bounds of the energy norm of the approximation error for linear elliptic partial differential systems. We generalize the complementarity error estimates known for scalar elliptic problems to general diffusion-convection-reaction linear elliptic systems. For systems we prove analogous properties of these error bounds as for the scalar case. A brief description how the presented general theory applies to linear elasticity is included as well as an application to chemical systems with reactions of at most first order. Numerical experiments showing the sharpness of the obtained upper bounds and their behavior in the adaptive procedure are presented, too. © 2012 Elsevier Inc. All rights reserved

The discrete maximum principle for Galerkin solutions of elliptic problems

This paper provides an equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green's function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-dimension and to the lowest-order finite elements on simplices of arbitrary dimension. The paper surveys the state of the art in the field of the discrete maximum principle and provides new generalizations of several results. © 2012 Versita Warsaw and Springer-Verlag Wien

Local a posteriori error estimator based on the hypercircle method

The error of the finite element solution of linear elliptic problems can be estimated a posteriori by the classical hypercircle method. This method gives accurate and guaranteed upper bound of the error measured in the energy norm. The disadvantage is that a global dual problem has to be solved, which is quite time-consuming. Combining the hypercircle method with the equilibrated residual method, we obtain locally computable guaranteed upper bound. The computer implementation of this a posteriori error estimator is also discussed