An initial-boundary value problem with a Caputo time derivative of fractional order α ∈ (0, 1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the L∞ and L 2 norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.1991 Mathematics Subject Classification. Primary 65M06, 65M15, 65M60.

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Abstract.We consider a deterministic model of landscape evolution through the mechanism of overland flow over an erodible substrate, using the St. Venant equations of hydraulics together with the Exner equation for hillslope erosion. A novelty in the model is the allowance for a nonzero bedload layer thickness, which is necessary to distinguish between transport limited and detachment limited sediment removal. It has long been known that transport limited uniform flow is unstable when the hillslope topography is geomorphologically concave (i.e., the center of curvature is above ground). In this paper, we show how finite amplitude development of the consequent channel flow leads to an evolution equation for its depth h of the form ht = h 3/2 + (h 3/2 ) Y Y , where Y is the cross-stream space variable. We show that solutions of compact support exist but that, despite appearances, blow up does not occur because of an associated integral constraint, and the channel equation admits a unique and apparently globally stable steady state. The consequences for hillslope evolution models are discussed.

We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the 'rising' side of the cylinder and, for large ones, pools at the cylinder's bottom. The paper examines rimming flows with a pool affected by weak surface tension. Using the lubrication approximation and the method of matched asymptotics, we find a solution describing the pool, the 'outer' region, and two transitional regions, one of which includes a variable (depending on the small parameter) number of asymptotic zones.

Residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polyhedral domains. Standard finite element approximations are considered. The error constants are independent of the diameters of mesh elements and the small perturbation parameter. In our analysis, we employ sharp bounds on the Green's function of the linearized differential operator. Numerical results are presented that support our theoretical findings.

A singularly perturbed quasi-linear two-point boundary value problem with an exponential boundary layer is discretized on arbitrary nonuniform meshes using first-and second-order difference schemes, including upwind schemes. We give first-and second-order maximum norm a posteriori error estimates that are based on difference derivatives of the numerical solution and hold true uniformly in the small parameter. Numerical experiments support the theoretical results.

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Abstract. A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green's function of the continuous differential operator in the Sobolev W 1,1 and W 2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.