A Dual Mesh Method for a Non-Local Thermistor Problem

We use a dual mesh numerical method to study a non-local parabolic problem arising from the well-known thermistor problem.


Introduction
In this work we propose a dual mesh numerical scheme for analysis of the following non-local parabolic problem coming from conservation law of electric charges: where ∇ denotes the gradient with respect to the x-variables. The nonlinear problem (1) is obtained, under some simplificative conditions, by reducing the well-known thermistor problem (cf., e.g., [13,14,15]), which consists of the heat equation, with joule heating as a source, and subject to current conservation: where the domain Ω ⊂ R 2 occupied by the thermistor is a bounded convex polygonal; ϕ = ϕ(x, t) and u = u(x, t) are, respectively, the distributions of the electric potential and the temperature in Ω; σ(u) and k(u) are, respectively, the temperature-dependant electrical and thermal conductivities; σ(u) |∇ϕ| 2 is the joule heating. The literature on problem (2) is vast (see e.g. [2,6,7,8,9,10,11,16,17]). With respect to numerical approximation results to problem (2) we are aware of [1,11,12,18]: in [18] a numerical analysis of the non-steady thermistor problem by a finite element method is discussed; in [12] the authors study a spatially and completely discrete finite element model; in [11] a semi-discretization by the backward Euler scheme is given for the special case k = Id; in [1] a box approximation scheme is presented and analyzed. A completely discrete scheme based on the backward Euler method with semi-implicit linearization to (2) is presented in [12] for the special case k(u) = 1. Existence and uniqueness of solutions to the problem (1) were proved in [10].
Finite volume methods emerged recently and seem to have a significant role on concrete applications, because they have very interesting properties in view of the subjacent physical problems: in particular in conservation of flows. An equation coming from a conservation law has a good chance to be correctly discretized by the finite volume method. We also recall that these schemes have been widely used to approximate solutions of the heat linear equation, semilinear or parabolic equations. Since we consider data f with lack of regularity when compared to previous work, we need a new way to discretize (1). We present a dual mesh method capable of handling the non local term λf (u) ( Ω f (u) dx) 2 which is a noticeable feature of (1), by generalizing the results of [1]. A box approximation scheme for discretizing (1) with the case k being different from the identity is obtained. Speed of convergence is directly related with regularity of the continuous problem. When one increases regularity of the second term and data, the solution see its regularity increasing in parallel, and precise speed of convergence can be established. In the existing literature (see e.g. [5,12]) the error estimates for both the finite element or volume element method are usually derived for solutions that are sufficiently smooth. Because the domain is polygonal, special attention has to be paid to regularity of the exact solution. We give sufficient conditions in terms of data and the solution u that yield error estimates (see hypothesis (H1) below).
The text is organized as follows. In Section 2 we set up the notation and the functional spaces used throughout the paper. Section 3 introduces a box scheme model for problem (1), and existence and uniqueness of the solution of the approximating problem (12) is obtained from the fixed point theorem and equivalence of norms in the finite dimensional space S 0 h . Finally, in Section 4, under some regularity assumptions, we prove error estimates.

Notation and functional spaces
Let (·, ·) and · denote the inner product and norm in L 2 (Ω); H 1 0 (Ω) = u ∈ H 1 (Ω), u/∂Ω = 0 ; · s , · s,p denote the H s (Ω) and the W s,p (Ω) norm respectively; T h denote a triangulation of Ω; T h v be the set of vertices of a quasi-uniform triangulation T h ; and S 0 h h>0 be the family of approximating subspaces of H 1 0 (Ω) defined by In the remainder of this paper we denote by c various constants that may depend on the data of the problem, and that are not necessarily the same at each occurrence. We assume that the family of triangulations is such that the following estimates [4] Let P h : L 2 (Ω) → S 0 h be the standard L 2 -projection. One has [4]: We construct the box scheme B h (dual mesh) employed in the discretization as follows. From a given triangle e ∈ T h , we choose a point q ∈ e as the intersection of the perpendicular bisectors of the three edges of e. Then, we connect q by straight-line segments to the edge midpoints of e.
To each vertex p ∈ T h v , we associate the box b p ∈ B h , consisting of the union of subregions which have p as a corner (see Fig. 1). For the piecewise constant interpolation operator I h , defined by we have the following standard error estimates [1,3]: We denote by N h (p) the set of the neighboring vertices of Fig. 1). Let l ∂b : ∂b → R + be defined as follows: w(x ± 0) are the outside and inside limit values of w(x) along the normal directions for ∂b. We now collect from the literature [1,3] some important lemmas and trace results, that are needed in the sequel.

Lemma 1. Assume that B h is a dual mesh. If v is a piecewise linear function, and x is not a vertex, then
where n is the unit outward normal vector on ∂b.
The h-dependent norms are defined as follows: and v 0,h = I h v .

Lemma 2.
There exists a constant c > 0 such that Moreover, if there exists a constant a 0 > 0 such that a ≥ a 0 in Ω, then where i h : C(Ω) → S 0 h is the Lagrangian interpolation operator and u ∈ H 2 (Ω).

Existence and uniqueness result for the box scheme method
Let u be the solution of (1). Integrating over an element b in B h we obtain: We consider a box scheme defined as follows: find u h ∈ S 0 h such that where u h (0) = P h u 0 and I h is the interpolation operator. Proof . We begin by proving existence of solution. We define a nonlinear operator G from S 0 h to S 0 h as follows. For each u h ∈ S 0 h , w h = G(u h ) is obtained as the unique solution of the following problem: We remark that G is well defined. Using v = w h as a test function in (13), hypotheses (H2) and (H3), and Holder's inequality, we can write: Thus, we have Integrating (14) with respect to t and using the equivalency of I h · and · in S 0 h (see (5)) yields Define now the following set We can easily see that D is closed subset of L ∞ (0, t, L 2 (Ω)) with its natural norm. We conclude that there exists t > 0 such that G(D) ⊂ D. To obtain that G has a fixed point w h = G(w h ), we prove that G is a contraction.
On the other hand, one has

Schwartz inequality implies that
By Lemma 5, we have Now, using v = w h 1 − w h 2 as a test function in (17), we obtain from (7): With use of the Holder's inequality and equivalency of I h · and · , integration of (18) with respect to time gives: Thus G is a contraction. We prove now uniqueness. Following the same arguments as before, we have Choosing v = u h 1 − u h 2 as test function in (19), using again (7) and integrating, we obtain which gives, by Gronwall's Lemma, uniqueness of solution.

Error analysis
In this section we prove error estimates under certain assumptions on regularity of the exact solution u.
Theorem 2. Under assumptions (H1)-(H4), if u, u h are solutions of (11)- (12) for 0 ≤ t ≤ t 0 (h), then Proof . From (11) and (12) we obtain We now estimate, separately, the terms on the right-hand side of (20). We have from (6) and (4) that By Lemma 5, inverse inequality (3) and (4), we have: Consequently, we obtain from (23) that Based on our earlier development in (16), we also know: Let v = u h − P h u be a test function in (20). Using Lemma 3, it follows from (21)-(25) that In order to estimate the right hand side of the last inequality, we treat both terms separately. By similar arguments to those used in (16), (27), by (9) and (10) we have Integrating (28), we arrive to and Theorem 2 gives On the other hand, by triangular inequality, (9) and the regularity of the exact solution u, we have We conclude then with the desired error estimate.

Conclusion
In this paper a dual mesh numerical scheme was proposed for a nonlocal thermistor problem.
We have showed the existence and uniqueness of the approximate solution via Banach's fixed point theorem. We have also proved H 1 -error bounds under minimal regularity assumptions.
We only obtain first-order estimates: higher order estimates are difficult to obtain due to the nonstandard nonlocal term. Optimal error analysis to the present context, under appropriate smoothness assumptions on data, can be derived by application of the techniques of [5], but this needs further developments.