We are concerned with the finite-element approximation for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. The scheme makes use of a semi-implicit time discretization with a time-increment control and Baba-Tabata’s conservative upwind finite-element approximation in order to realize the positivity and mass conservation properties....

We examine the use of orthogonal spline collocation for the semi-discretization of the cubic Schr odinger equation and the two-dimensional parabolic equation of Tappert. In each case, an optimal order L 2 estimate of the error in the semidiscrete approximation is derived. For the cubic Schr odinger equation, we present the results of numerical experiments in which the integration in time is p...

We derive residual-based a posteriori error estimates of optimal order for time discretizations by the fractional-step θ-scheme for linear parabolic equations. First, we consider the time semi-discrete problem. The main tool of our analysis is an appropriate reconstruction of the piecewise linear interpolant of the approximate solution that leads to a residual of optimal order. Next, we extend ...

In this paper, we present a highly accurate Hamiltonian structure-preserving numerical method for simulating Hamiltonian wave equations. This method is obtained by applying spectral variational integrators (SVI) to the system of Hamiltonian ODEs which are derived from the spatial semi-discretization of the Hamiltonian PDE. The spatial variable is discretized by using high-order symmetric finite...

Stabilized semi-implicit spectral defect correction (SSISDC) methods are constructed for the time discretization of Allen-Cahn and Cahn-Hilliard equations. These methods are unconditionally stable, lead to simple linear system to solve at each iteration and can achieve high-order time accuracy with a few iterations in each time step. Ample numerical results are presented to demonstrate the effe...

We provide error analyses for explicit, implicit, and semi-implicit monotone finitedifference schemes on uniform meshes with nonlocal numerical fluxes. We are motivated by finite-difference discretizations of certain long-wave (Sobolev) regularizations of the conservation laws that explicitly add a dispersive term as well as a nonlinear dissipative term. We also develop certain relationships be...

In this paper we discuss the numerical analysis of an upscaled (core scale) model describing the transport, precipitation and dissolution of solutes in a porous medium. The particularity lies in the modeling of the reaction term, especially the dissolution term,which has amultivalued character.We consider theweak formulation for the upscaled equation and provide rigorous stability and convergen...

In this acticle a semi-smooth Newton method for frictional two-body contact problems and a solution algorithm for the resulting sequence of linear systems is presented. It is based on a mixed variational formulation of the problem and a discretization by finite elements of higher-order. General friction laws depending on the normal stresses and elasto-plastic material behaviour with linear isot...

We consider the numerical solution of a nonlinear evolutionary variational inequality, arising in the study of quasistatic contact problems. We study spatially semi-discrete and fully discrete schemes for the problem with several discontinuous Galerkin discretizations in space and finite difference discretization in time. Under appropriate regularity assumptions on the solution, a unified error...

Combining the numerical concept of variational discretization introduced in [11, 12] and semi-smooth Newton methods for the numerical solution of pde constrained optimization with control constraints [9, 22] we place special emphasis on the implementation and globalization of the numerical algorithm. We prove fast local convergence of a globalized algorithm and illustrate our analytical and alg...