Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor

A Finite Element Model for the Time-Dependent Joule Heating Problem

“…The existence of global weak solutions in two space dimensions (il c R2) was shown by Cimatti [5], see also Rodrigues [15] and Allegretto and Xie [2] for existence results for related problems. The regularity of these solutions, however, is insufficient for the purpose of proving error bounds of optimal order, cf.…”

“…Theorems 3.1 and 3.3. Building upon the techniques of [5], we obtain global strong solutions with the required regularity in one and two space dimensions. The three-dimensional case remains an open problem.…”

“…Finally, in §4 we prove the global existence of solutions with the required regularity when d < 2. Our argument here builds upon the techniques of Cimatti [5], who showed the existence of weak solutions. We are not aware of any existence and regularity result in the three-dimensional case.…”

“…In particular, we avoid using a maximum norm bound for V</>/,, which would be difficult to obtain. We also consider a completely discrete scheme based on the backward Euler method with semi-implicit linearization: find Un £ Sh , ®n € Sh with <!>" -nhg(tn) £ Sh such that (BnUn , X) + (Vt/", V*) = (o(Un-X)\V<!>n-X\2, X), (1)(2)(3)(4)(5)(6)(7)(8) Vxe$k, tn€(0,T], Uo = Uho , and (1.9) (cr(c7n)V<D",Vy) = 0, V* £ Sh, tn £ [0, T).…”

Existence and uniqueness for the nonstationary problem of the electrical heating of a conductor due to the Joule‐Thomson effect

“…(s) _< ,s e R for some M _> m the existence of a weak solution is established for (1.1) in [2]. A result due to Shi, Shillor, and Xu [3] asserts that the assumption that N 2 and a cl(tt) in [2] can be eliminated.…”