Modelling of mantle flows with sharp viscosity contrasts in a
viscoelastic medium is a challenging computational problem in
geodynamics because of its multiple-scale nature in space and time. We
have employed a recently developed adaptive multilevel wavelet
collocation algorithm to study the dynamics of a small rising diapir
interacting with a stiff lithosphere in a Maxwell viscoelastic mantle.
In this kinematic model we have prescribed the upward velocity of the
diapir and then we need to integrate in time only the momentum equation
governing the temporal evolution of the pressure, stress and velocity
components, which together constitute a sixth-order system in time. The
total number of collocation points did not exceed 10(4), compared to
more than 10(6) gridpoints using conventional evenly spaced grid
methods. The viscosity of the diapir is 10(-4) times lower than that of
the surrounding mantle, while the viscosity of the thin lithosphere,
about 5-10 per cent of the entire layer depth, is 10(4)-10(8) times
stiffer than the ambient mantle. Our results demonstrate the efficacy of
wavelets to capture the sharp gradients of the stress and pressure
fields developed in the diapiric impingement process. The interaction of
the viscoelastic lithosphere with the rising viscoelastic diapir results
in the localization of stress within the lithosphere. The magnitude of
the stress fields can reach around 100-300 MPa. Our simple kinematic
model shows clearly that viscoelasticity can potentially play an
important role in the dynamics of the lithosphere, especially concerning
the potential severage of the lithosphere by mantle upwellings.