The Numerical Study On Some Stochastic Partial Differential Equation

Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using these so-called stochastic partial differential equations(SPDEs) is that SPDEs are able to more fully capture the behavior of interesting phenomena; it also means that the corresponding numerical analysis of the model will require new tools to model the systems,produce the solutions,and analyze the information stored within the solutions.In this thesis, we consider two different stochastic partial differential equations.Firstly, we study finite element methods for the boundary value problem of a semilinear stochastic elliptic partial differential equation driven by an additive white noise,and then we show several fast slovers for the numerical solutions of finite element methods with color noise.Consider following semilinear stochastic elliptic partial differential equation whereΩis a bounded convex domain in Rd(d=2,3) with smooth boundary,f and g are given deterministic function and W denotes the white noise. Here,we shall assume that f satisfies the following conditions:(A1) There is a constantα<γsuch that whereγis the positive constant in the Poincare's inequality (see [1]).(A2) f is Lipschitz continuous,i.e., there is a positive constant Lq such that Theorem 1. ([12]) If f satisfies (A1) and (A2), u and uhs is the solution of(4.5) and FEM respectively. Then where C is constant independent on h.When W denotes the color noise, we discuss two kinds of fast solver:Poisson-Schwarz-Newton method and Newton-Krylov-Schwarz method.We have following two convergence theorems.Theorem 2.([51]) If we choose 0<λ<2β1/β2, whereβ1 andβ2 are two parame-ters,then Poisson-Schwarz-Newton method converges in the sense that Hereμ= 1-λβ2(2β1/β2-λ)<1 and C are independent of the mesh parameters h and H.Furthermore,we haveTheorem 3.([51])There exist constants c1 and c2 both sufficiently small, such that andρm≤c2 for all m,then Newton Krylov Schwarz method converges in the sense that Here 0≤μ＜1 is a constant independent of the mesh parameters.Furthermore, we haveSecondly, we study stochastic Helmholtz equation driven by an additive white noise,with a bounded convex domain with smooth boundary of Rd(d=2,3).And then with the help of the perfectly matched layers technique,we also consider the stochastic scattering problem of Helmholtz type.We study the following stochastic Helmholtz equation driven by an additive white noise forcing term: whereΩis a bounded convex domain in Rd(d=2,3) with smooth boundary,f is a given deterministic function and W denotes the white noise.To simplify our presentation we assume that the coefficient of the white noiseσ(x)=1.Also we assume throughout the paper that the wave-number k is positive and bounded away from zero,i.e.,k≥k0＞0.We use FEM and DG for wave-number k is low and middle respectively, and we have following two convergence theorems:Theorem 4.([14])LetΩbe a bounded convex domain with smooth boundary,u and uhs be the solution of (3.1)and FEM respectively. If the mesh satisfies hk2(?)1 and k2 is not an eigenvalue of -Δ, then we have where C is a positive constant independent of u and h.Theorem 5. ([14]) LetΩbe a bounded convex domain with smooth boundary, u and uhs be the solution of (4.6) and DG respectively. If the mesh satisfies hk2(1+ hk)(?)1 and h sufficiently small, then where C＞0 is a constant independent of u and h.We also propose and study an perfectly matched layer (PML)technique for solving exterior Helmholtz problems with perfectly conducting boundary: HereΩis a bounded convex domain with smooth boundary of Rd(d=2,3), g∈H-1/2((?)Ω) is determined by the incoming wave, and n is the unit outer normal to (?)Ω.We assume the source term have compact support belong inΩ1\Ω(Ω(?)Ω1), i.e. supp(f+W) (?)Ω1\Ω.We use FEM and DG for wave-number k is low and middle respectively, and we have following two convergence theorems:Theorem 6.([13]) AssumeΩR be a bounded domain with smooth boundary,and u and uP,hs be the solution of (4.7) and FEM respectively. Let the mesh satisfies hk2(?)1 and k2 is not an exterior eigenvalue.Then for sufficiently largeσ0>0,we have where C is a positive constant independent of h.Theorem 7.([13]) Let u and uP,hs be the solution of (4.7) and DG respectively. If the mesh satisfies hk2(1+hk)> 1 and h small enough, we have with C>0 independent of h and k.