Volume 23, Issue 2

In this paper we construct a numerical solver for the Saint Venant equations. Special attention is given to the balancing of the source terms, including the bottom slope and variable cross-sectional profiles. Therefore a special discretization of the pressure law is used, in order to transfer analytical properties to the numerical method. Based on this approximation awell-balanced solver is developed, assuring the C-property and depth positivity. The performance of this method is studied in several test cases focusing on accurate capturing of steady states.

We consider finite difference solutions of the 1-dimconvective reaction-diffusion equation u t = u xx + ɑ(u m ) x +u β (ɑ > 0, β > m ≥ 1),whose solutions are known to become unbounded in finite time. To reproduce the finite-time blow-up phenomenon numerically, the temporal grid sizes are defined adaptively.We shownot only that our numerical solution blows up in finite time but also that the numerical blow-up time converges to the blow-up time of the PDE. In addition, we also investigate the behavior of the numerical solutions. Our numerical solutions reproduce certain significant blow-up properties of the continuous solutions.

Various applications in computational fluid dynamics and solid mechanics motivate the development of reliable and efficient adaptive algorithms for nonstandard finite element methods (FEMs). Standard adaptive finite element algorithms consist of the iterative loop of the basic steps Solve, Estimate, Mark and Refine. For separate marking strategies, this standard scheme may be universalised. The (total) error estimator is split into a volume term and an error estimator term. Since the volume term is independent of the discrete solution, an appropriate data approximation may be realised by a high degree of local mesh refinement. This observation results in a natural adaptive algorithm based on separate marking. Its quasi-optimal convergence is proven in this first part for the Poisson model problem and nonconforming Crouzeix-Raviart FEM. The numerical experiments confirm the optimal convergence rates and reveal its flexibility. The second part is devoted to the quasi-optimality of S-AFEM-AA and the pure displacement problem in linear elasticity and the Stokes equations.

Various applications in computational fluid dynamics and solid mechanics motivate the development of reliable and efficient adaptive algorithms for nonstandard finite element methods (FEMs). Standard adaptive finite element algorithms consist of the iterative loop of the basic steps Solve, Estimate, Mark, and Refine. For separate marking strategies, this standard scheme may be universalised. The (total) error estimator is split into a volume term and an error estimator term. Since the volume term is independent of the discrete solution, an appropriate data approximation may be realised by a high degree of local mesh refinement. This observation results in a natural adaptive algorithm based on separate marking. Its quasi-optimal convergence is proven in this second part for the pure displacement problem in linear elasticity and the Stokes equations and nonconforming Crouzeix-Raviart FEM. The proofs follow the same general methodology as for the Poisson model problem in the first part of this series. The numerical experiments confirm the optimal convergence rates and reveal its flexibility.

The paper is concerned with the superconvergence of numerical evaluation of Hadamard finite-part integral. Following the works [6-9], we studied the second-order and the third-order quadrature formulae of Newton-Cotes type and introduced new rules. The rule for the second-order gives the same convergence rate as the rule [6] but in more general cases, the rule for the third-order gives better results than the rule in [9] In this work, first we mention the main results on the superconvergence of the Newton-Cotes rules, we mention trapezoidal and Simpson’s rules and then we introduce a rule based on the cubic approximation. In the second part we describe important error estimates and in the last section we demonstrate theoretical results by numerical examples.