Co-volume methods for degenerate parabolic problems. (English) Zbl 0797.65075
A completely volume (co-volume) technique is used for developing a physically appealing algorithm for the solution of degenerate parabolic problems such as for example a Stefan problem which models heat conduction in a material which may undergo a change in phase. In each of phases, the balance of energy reduces to the classical heat conduction problem, \(de/dt - k_ i \Delta u = f\) in \(\Omega_ i\), where \(e\) is the energy, \(u\) the temperature and \(k_ i\) the conductivity in phase \(i\) which currently occupies a region \(\Omega_ i \subset \Omega\). In each phase, \(u\) is typically an affine function of the energy \(e\) with the slope \(c_ i\), the specific heat of the \(i\)th phase; however, any monotone function of \(e\) may be accomodated.
It is shown that the suggested algorithms give rise to a discrete semigroup theory that parallels the continuous problem. In particular, the discrete Stefan problem gives rise to a nonlinear semigroup in both the discrete \(L^ 1\) and \(H^{-1}\) spaces.
It is shown that the suggested algorithms give rise to a discrete semigroup theory that parallels the continuous problem. In particular, the discrete Stefan problem gives rise to a nonlinear semigroup in both the discrete \(L^ 1\) and \(H^{-1}\) spaces.
Reviewer: J.Vaníček (Praha)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65Y05 | Parallel numerical computation |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 80A22 | Stefan problems, phase changes, etc. |
| 35K55 | Nonlinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 20M99 | Semigroups |
Keywords:
co-volume methods; phase change; parallel computation; degenerate parabolic problems; Stefan problem; heat conduction; discrete semigroup theory; nonlinear semigroupReferences:
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