a b s t r a c tThis article presents a splitting technique for solving the time dependent incompressible Navier-Stokes equations. Using nested finite element spaces which can be interpreted as a postprocessing step the splitting method is of more than second order accuracy in time. The integration of adaptive methods in space and time in the splitting are discussed. In this algorithm, a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm. Results on the 'Flow around a cylinder's-and the 'Driven Cavity's-problem are presented.
Abstract. The numerical solution of time-dependent partial integro-differential equations is considered. This includes both variants, the initial value problem and the prehistory problem. One problem concerning partial integro-differential equations is the data storage. To deal with this problem an adaptive method of third order in time is developed to save storage data at smooth parts of the solution. Beyond this a post-processing step adaptively thins out the history data.
A numerical method for a time-dependent nonlinear partial integro-differential equation (PIDE) is considered. This PIDE describes a spatial population model that includes a given carrying capacity and the memory effect of this environment. To deal with this issue an adaptive method of third order in time is considered to save storage data in smooth parts of the solution. Beyond this, a post-processing step adaptively thins out the history data.
The numerical solution of a parabolic convection diffusion equation with delay term is considered. This includes both variants, the initial value problem and the prehistory problem. Equations with a delay or memory term, often called integrodifferential problems, appear in different contexts of heat conduction in materials with memory, viscoelasticity and population models. This work concentrates on the linear convection diffusion case of the prehistory and the initial value problem. One problem concerning delay or memory problems is the data storage. To deal with this problem an adaptivity method of third order in time is developed to save storage data at smooth parts of the solution. Numerical results for higher Péclet numbers are presented.
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