Error analysis for function representation by the sparse-grid combination technique

Detailed error analyses are given for sparse-grid function representations through the combination technique. Two- and three-dimensional, and smooth and discontinuous functions are considered, as well as piecewise-constant and piecewise-linear interpolation techniques. Where appropriate, the results of the analyses are verified in numerical experiments. Instead of the common vertex-based function representation, cell-centered function representation is considered. Explicit, pointwise error expressions for the representation error are given, rather than order estimates. The paper contributes to the theory of sparse-grid techniques

The sparse-grid combination technique applied to time-dependent advection problems

In the numerical technique considered in this paper, time-stepping is performed on a set of semi-coarsened space grids. At given time levels the solutions on the different space grids are combined to obtain the asymptotic convergence of a single, fine uniform grid. We present error estimates for the two-dimensional spatially constant-coefficient model problem and discuss numerical examples. A spatially variable-coefficient problem (Molenkamp-Crowley test) is used to assess the practical merits of the technique. The combination technique is shown to be more efficient than the single-grid approach, yet for the Molenkamp-Crowley test, standard Richardson extrapolation is still more efficient than the combination technique. However, parallelization is expected to significantly improve the combination technique's performance

Numerical solution of mixed gradient-diffusion equations modelling axon growth

In the current paper a numerical approach is presented for solving a system of coupled gradient-diffusion equations which acts as a first model for the growth of axons in brain tissue. The presented approach can be applied to a much wider range of problems, but we focus on the axon growth problem. In our approach time stepping is performed with a Rosenbrock solver with approximate matrix factorization. For the Jacobian an approximation is used that simplifies the solution of the coupled parabolic and gradient equations. A possible complication in the implementation of source terms is noted and a criterion that helps to avoid it is presented