Abstract
In this paper we consider one-dimensional partial differential equations of parabolic type involving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition. We prove existence and uniqueness result by stochastic methods which also allow us to develop a low complexity Monte Carlo numerical resolution method. We get accurate pointwise estimates for the derivatives of the solutionfrom which we get sharp convergence rate estimates for our stochastic numerical method.
Highlights
Given a finite time horizon T and a positive matrix-valued function a(x) which is smooth except at the interface surfaces between subdomains of Rd, consider the parabolic diffraction problem ∂tu(t, x) −1 2 div(a(x)∇)u(t, x) = for all (t, x) ∈ (0, T ] ×Rd, u(0, x) = f (x) for all x ∈ Rd
As stated in Ladyzenskaja et al [12, chap.III, thm.13.1] 1, there exists a unique solution u(t, x) to (1) with compatibility transmission conditions belonging to the space V21,1/2([0, T ] × Rd); this solution is continuous, twice continuously differentiable in space and once continuously differentiable in time on (0, T ] × (Rd − ∪iSi)
For the sake of completeness and because of its importance in our analysis, we will prove this existence and uniqueness theorem in the one-dimensional case by using stochastic arguments essentially; this approach allows us to get the precise pointwise estimates on partial derivatives of u(t, x) which are necessary to get sharp convergence rate estimates for our transformed Euler scheme
Summary
Given a finite time horizon T and a positive matrix-valued function a(x) which is smooth except at the interface surfaces between subdomains of Rd, consider the parabolic diffraction problem. Our first objective is to provide a probabilistic interpretation of the solutions which allows us to get pointwise estimates for partial derivatives of the solution u(t, x) These estimates, which are interesting in their own, allow us to complete our second objective, that is, to develop an efficient stochastic numerical approximation method of this solution and to get sharp convergence rate estimates. The positive real numbers denoted by C may vary from line to line; they only depend on the functions f and σ, the point x0, and the time horizon T
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