Resolving singularities in parabolic initial-boundary value problems

Torsten Linß (FernUniversität in Hagen), Brice Girol

We consider a time-dependent reaction-diffusion equation with a singularity arising from incompatible initial and boundary conditions: 2utuxx+b(x,t)u=fin  (0,)×(0,T],subject to boundary conditionsu(0,t)=φ0(t),   u(,t)=φ(t),t(0,T],and the initial conditionu(x,0)=0,x(0,), with φ0(0)0.

The discrepancy between initial and boundary conditions causes the formation of a singularity in the vicinity of the corner (0,0). This singularity s can be characterised as the solution of 2stsxx+b(0,0)s=0in  (0,)×(0,T],subject to the boundary conditions(0,t)=φ0(0),t(0,T],and the initial conditions(x,0)=0,x(0,). This in turn can be given analytically using the error function. Then the interesting question is:

How can the remainder y=us be resolved numerically?

We derive bounds on the derivatives of the remainder y — under significantly less restrictive assumptions then previously assumed by other authors — and show how a numerical approximation can be obtained using an appropriately designed mesh.

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