- Open Access
Iterative methods for ternary diffusions
© Leszczyński and Wrzosek; licensee Springer. 2014
Received: 31 January 2014
Accepted: 9 April 2014
Published: 5 May 2014
We apply iterative methods to three-component diffusion equations and study theirconvergence in and in the Sobolev space . The system is parabolic and mass-conservative.Newton’s method converges very fast and its iterations do not leave theset of admissible functions.
MSC: 35K51, 35K57, 65M12, 65M80.
Since its discovery and later analysis by Darken , the Kirkendall effect  has been found in various alloy systems, and studies on lattice defectsand diffusion developed significantly. The Danielewski-Holly method  extends the Darken standard theory of interdiffusion and describes theprocess in the bounded mixture showing constant concentration. Under certainregularity assumptions and quantitative condition Danielewski and Holly proved theexistence and uniqueness of solution to PDE describing the interdiffusion phenomena.Further developments have been presented in numerous articles; e.g.[4, 5].
In the paper we apply Newton’s method (see [6–8]) to three-component diffusion equations and study the convergence in and Sobolev space . The system of equations is strongly coupled,however, the maximum principle presented in Section 1 confirms its parabolictype. Parabolicity is additionally confirmed by our convergence result for iterativemethods. This falsifies the nonparabolicity hypothesis by Danielewski and Holly , where they construct an initial concentration whose norm increases in time, at least on some interval.The Newton method, known as quasilinearization method, is very useful inmodern numerical methods for solving PDE’s; see . We apply this method to strongly coupled parabolic systems describingdiffusing mixtures. This strong parabolicity might have caused weird phenomena, butwe have discovered a kind of maximum principle and some conservation laws in thissystem, hence the iterative methods proposed here behave very well. Our result isvery useful in numerical simulations when one wants to construct reliable and fastconvergent approximations. Since Newton’s method produces linear PDE’ssatisfying maximum principles and a priori estimates of the respectiveGreen functions or Cauchy kernels, one can find errors estimates much better thanthose obtained from the Newton-Kantorovich theorem, cf. [10, 11].
, , , are bounded,
, , for , ,
for , ,
u, v, w obey the Neumann boundarycondition.
Remark 1.1 If then the third equation of (1.1) is not necessary,since . However, we keep it for a more convenient analysisof some properties of solutions.
the drift velocity; it describes the marker position.
Lemma 1.3 (Mass conservation)
can be shown by means of the Neumann boundary condition. □
Lemma 1.4 (Maximum principle)
for. Ifsatisfy (1.1)-(1.3) then.
which is a contradiction. Thus for , . If then . Hence . Similarly, and for , . □
Let be the closure of w.r.t. the norm. The existence and uniqueness of solutions toproblem (1.1)-(1.3) in w.r.t. the Sobolev norm is given in . The following proposition concerns the uniqueness of solutions in. Since the set of -functions is dense in , the proof is carried out in . The uniqueness isobtained for weak solutions.
Proposition 2.1 Assume thatand. Then a weak solutionto problem (1.1)-(1.3) is unique in.
implies for all admissible , . □
3 Iterative methods
Lemma 3.1 Assume, atand. Iffulfills (3.1) with (3.2), the Neumann boundary conditionand (3.3), then.
Hence the statement is proved. □
The following theorem establishes a convergence of the iterative method(3.1)-(3.2).
then the sequencedefined by (3.1), (3.2) converges to the solutionof (1.1), (1.2) in the Sobolev space.
By the d’Alembert’s ratio test the convergence radius is+∞. □
Remark 3.4 The functions can be slightly perturbed near the lateral boundaryin order to fulfill the Neumann boundary condition.
4 Convergence of the Newton method
with the initial condition (3.2) and the Neumann boundary condition.
Lemma 4.1 Assume, atand. Iffulfills (4.1) with (3.2) and the Neumann boundarycondition, then.
is . □
The following theorem establishes the convergence of the Newton method.
then the sequencedefined by (4.1), (3.2) converges to the solutionof (1.1)-(1.3) with respect to the norms in the Sobolev space.
This example of iterations shows that strongly coupled systems cause seriousproblems with their approximation. We think that the ternary system and suitableapproximations to it will be somehow expressed in an abstract way, based on aConti-Opial type theorem, like in .
Maximal differences between successive approximations by direct iterations ( 3.1 ) with , , , ,
Maximal differences between successive approximations by Newton’s method ( 4.1 ) with , , , ,
This work was supported by the National Science Center (Poland) decision no.DEC-2011/02/A/ST8/00280.
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