Volume 30, Issue 6

We consider an initial-boundary value problem for a sixth order Cahn-Hilliard equation describing the formation of microemulsions. Based on a Ciarlet-Raviart type mixed formulation as a system consisting of a second order and a fourth order equation, the spatial discretization is done by a C 0 Interior Penalty Discontinuous Galerkin (C 0 IPDG) approximation with respect to a geometrically conforming simplicial triangulation of the computational domain. The DG trial spaces are constructed by C 0 conforming Lagrangian finite elements of polynomial degree k ≥ 2. This leads to an initial value problem for an index 1 Differential Algebraic Equation (DAE) which is further discretized in time by an s-stage Diagonally Implicit Runge-Kutta (DIRK) method of order p ≥ 2. The resulting parameter dependent nonlinear algebraic system is solved numerically by a predictor-corrector continuation strategy with constant continuation as a predictor and New- ton’s method as a corrector featuring an adaptive choice of the continuation parameter. Numerical results illustrate the performance of the suggested approach.

The paper suggests a preconditioning type method for fast solving of elliptic equations with oscillating quasiperiodic coefficients A ε specified by the small parameter ε > 0. We use an iteration method generated by an elliptic operator, associated with a certain simplified (e.g., homogenized) problem. On each step of this procedure it is required to solve an auxiliary elliptic boundary value problem with non-oscillating coefficients, where typically the coefficients are smooth or piecewise constant. All the information related to complicated coefficients of the original differential problem is encompasses in the linear functional, which forms the right hand side of the auxiliary problem. For this reason, inversion of the original operator associated with oscillating coefficients is avoided. The only operation required instead is multiplication of it on a vector (vector function), which can be efficiently performed due to the low QTT-rank tensor operations with the rank parameter controlled by the given precision δ > 0 independent on the parameter ε. We prove that solutions generated by the iteration method converge to the solution of the original problem provided that the parameter of the iteration algorithm has been properly selected. Moreover, we deduce two-sided a posteriori error estimates that do not use A ε −1 and enable us to compute guaranteed bounds of the distance to the exact solution of the original problem for any step of the iteration process. For a wide class of oscillating coefficients, we obtain sharp QTT rank estimates for the stiffness matrix in tensor representation. In practice, this leads to the logarithmic complexity scaling of the approximation and solution process in both the FEM grid-size, and the frequency parameter 1/ε. Numerical tests in 1D confirm the logarithmic complexity O(|log ε |) of the proposed method applied to a class of complicated highly-oscillating coefficients.

Spatial motion of an ideal inextensible thread is described by six first order partial differential equations with one space variable and one algebraic equation that has to be valid at each point. In the paper, new formulation of the problem without an algebraic equation is suggested. It is proved that the new problem is equivalent to the original one and an efficient numerical method for its solution is developed.

We consider an optimal control problem of a system governed by a linear parabolic equation with the following features: control is distributed, observation is either distributed or final, there are constraints on the state function and on its time derivative. Iterative solution methods are proposed and investigated for the finite difference approximations of these optimal control problems. Due to explicit in time approximation of the state equation and the appropriate choice of the preconditioners in the iterative methods, the implementation of all constructed methods is carried out by explicit formulae. Computational experiments confirm the theoretical results.

The paper develops a semi-Lagrangian method for the numerical integration of the transport equation discretized on adaptive Cartesian cubic meshes. We use dynamically adaptive graded Cartesian grids. They allow for a fast grid reconstruction in the course of numerical integration. The suggested semi- Lagrangian method uses a higher order interpolation with a limiting strategy and a back-and-forth correction of the numerical solution. The interpolation operators have compact nodal stencils. In a series of experiments with dynamically adapted meshes, we demonstrate that the method has at least the second-order convergence and acceptable conservation and monotonicity properties.