Numerical Methods For Several Types Of Stochastic Partial Differential Equations

In this paper,we focus on three types of stochastic partial differential equations(SPDEs),which consist of the random Brinkman problem,the random interface Helmholtz problem,and the random interface Maxwell problem.We design the algorithms corre-sponding to the problems,prove their convergence theoretically,and give numerical ex-periments to verify the efficiency.This paper is mainly divided into the following five parts:In Chapter 1,we review the development of the finite element method,and the histo-ry of the weak Galerkin finite element method and the Nedelec element.Then the random Brinkman problem,the random interface Helmholtz problem and the random interface Maxwell problem are briefly summarized.In the end of this chapter,we give the algo-rithms of this thesis.In Chapter 2,we mainly study the random Brinkman problem.The Brinkman model of porous media,which is a generalization of the Stoke's equations and an approximation of the Navier-Stokes equations at low Reynolds numbers,behaves like a Darcy flow and a Stokes flow for the regions with large and small permeability values,respectively.The Brinkman model is a combination of Darcy's and Stoke's equations,which is a very effective model for simulating flows in highly heterogeneous media in real applications.Therefore.the accuracy of the Brinkman flow simulation is of significant practical interest,but it is not easy to design uniform stable algorithms to capture the behavior of Darcy flow and Stokes flow in different permeability regions simultaneously.Generally speaking,the usual Darcy stable method does not work well for Stoke flow and vice versa.Based on the weak gradient operator,Mu,Wang,and Ye proposed weak Galerkin scheme for the deterministic Brinkman equation recently,which is uniformly stable for large and small permeability regions.The most important advantage of this method is that one formulation works well for both the Darcy and the Stokes problems.In many practical applications,the interfaces of Darcy and Stokes domain are un-known beforehand,which is equivalent to that the permeabilities are random variables or the fluid viscosities coefficients jumps stochastically.The stochastic partial differential equations(SPDE)is a powerful tool which adequately describes the behavior of flows in highly heterogeneous media with random permeability and viscosity.Let D(?)R2 be a bounded convex domain with piecewise smooth boundary and(?,F,P),be a probability space.Here,we shall consider the following random Brinkman equation:find the velocity u(?).and the pressure p(?) such that(?)where the viscosity,is a stochastic function with jump and ? is the stochastic permeabil-ity tensor.Let the domain D be the union of Darcy's region(denoted by Dd)and Stoke's region(denoted by Ds)which have substantially different viscosity ?d>0 and ?s>0,respectively,that is D=Dd ? Ds.These two regions have different viscosity coefficients,denoted by ?d>0 and ?s>0.The random viscosity ?(x,?)in(1)?(3)is considered as a random jump coefficient.Here,we adopt the Levy processes method to reconstruct two phase composite materials,according to the given samples or given statistical characteris-tics.The forcing term f ? L2(D)2 and the boundary data g ?H1/2((?)D)2 are deterministic functions with the compatibility conditionThis type of random problems have many applications in industrial and engineering phe-nomena,such as groundwater systems,vuggy porous media,etc.For the sake of simplicity,we consider g = 0 in the sequel.We will employ an efficient MLMCWG method for solving random Brinkman equa-tion(1)?(3).There are several merits of our algorithms.As mentioned above,one of themain advantages of the random Brinkman model is that it can capture Stokes and Darcy type flow behavior depending on the value of ? without priori information of the interface,and the WG method is uniformly stable for Darcy's and Stoke's regions for each realiza-tion of(1)?(3).Secondly,the multi-level Monte Carlo technique has been widely used to replace traditional MC-like methods,so that the computational cost can be sharply re-duced.Finally,the interfaces between Darcy's and Stoke's regions are always complicated,and the traditional triangular partitions may not be suitable for practical computations.On the other hand,the WG method allows arbitrary polygons as long as the partitions are shape regular which is more efficient than standard finite element method(FEM),and the corresponding MLMCWG is superior than MLMCFEM for random case.The convergence analysis of WG method with different polygons are guaranteed under the same framework,which makes this method more flexible and robust.We're at the stage to present the H1-like convergence result of MLMCWG method.Theorem 1 Let(u,p)?L2([H01)? Hk+1(D)]2;?)×L2((L02(D)?Hk(D));?)and(UhN,phN)?UhN×PhN be the solutions of(2.1)?(2.3)and MLMCWG method respec-tively.Then,we have the following error estimate where the constant C is independent of hn and Mn.The numerical simulations are shown to verify the efficiency and robustness of the MLMCWG method.In Chapter 3,we mainly study the random interface Helmholtz problem.Diffraction gratings are the optic elements with periodic structures,which have important applicationsin industrial and engineering fields.Among these remarkable methods,electromagnetic wave diffraction governed by the Maxwell equation in the diffraction grating region is an efficient and popular tool nowadays.The rigorous comput,ation of the time-harmonic Maxwell equation diffracted by one dimensional periodic material can be reduced to the two dimensional Helmholtz equation with associated boundary conditions.The surfaces of interfaces,particularly the instruments in nanoscale,are far from perfectly smooth which can result in an increase of error between design and production.Though these variations may not be truly random,they can often be accurately modeled by assuming random surface roughness with an appropriate spatial correlation.To keep our presentation as elementary as possible,we confine ourselves to one dimensional stochastic interface grating problems in this work,in which the boundary is described by the graph of a random function.In this work,we propose an efficient numerical method for solving the random inter-face grating problem based on the shape derivatives,the weak Galerkin method,and the pivoted Cholesky decomposition(PCD)techniques.First,the following lemma presents the first order shape derivative du or the interface problem with respect to the nominal interface ?0.Lemma 2 Assume the deterministic perturbed interface in(3.31)satisfies ?? ? C2,1(?0,RN),and ?0 is sufficiently small to ensure that the interface ?? is not degenerate and lies still inside the domain D0.Then the first order shape derivative du exists and satisfies the following interface problem (?)Using the standard shape Taylor expansion technique,we can obtain the following results of approximation for the expectation Eu(x).Theorem 3 Let u and u be the solutions of the stochastic interface grating problem(3.14)and the deterministic interface grating problem(3.17)?(3.23)respectively.Under the assumptions of Theorem 3.3,then the following first order shape Taylor expansion holds (?)for all x ? K,a.s.? ? ?.Moreover,the following approximation of the expectation Eu(x)holdsTo avoid solving(3.51)directly,we shall use an efficient low-rank approximation based on the pivoted Cholesky decomposition to approximate Cordu(x,y)which will be specified in the next section.Once with the quantity Cordu(x,y)at hand,using the identity(4.51),we can approximate the variance Varu(x)by the following Theorem.Theorem 4 Under the assumptions of Theorem 3.3,then the following estimate holds for the variance Varu(x)of the solution of random interface grating problem(3.14)as Var(?) is the solution to(3.51).The solution itself or its gradient,or its each realization changes rapidly near the interface,and thus the traditional finite element method cannot capturing the jump or oscillations efficiently.Here,we adopt the weak Galerkin methods(WGMs)to conquer this issue,which use polynomials and thus high-order accuracy and high resolution can be obtained in regions where the solution is smooth and in the neighborhood of oscillations,respectively.In order to verify the theoretical estimates for expectation and variance of stochastic solution,in terms of second and third orders of the perturbation magnitude re-spectively,we also use the multi-level Monte Carlo weak Galerkin method(MLMCWGM)for the random grating problem to obtain the reference expectation and variance.Finally,the numerical simulations are shown to verify the efficiency and robustness of method.In Chapter 4,we mainly study the random interface Maxwell problem.Electromagnetic wave interacting with physical objects has important applications in industrial and engineering fields,such as wide band antennas,telecommunication chips,and remote sensing,etc.Electromagnetic fields have singularities or oscillations not only at the corner and edges of the computational domain,but also the interfaces between the different materials.Many mathematical models governed by Maxwell's equations are pro-posed for practical purposes,for example equations with materials interfaces,equations with discontinuous coefficients.In many practical applications,the interfaces of differ-ent materials are unknown beforehand,which is equivalent to that the permittivity and permeability are random variables.This chapter is a generalization of the previous chapter for three-dimensional case.In this chapter,we also propose an efficient numerical method for solving random inter-face interface Maxwell problem based on the shape derivatives and the pivoted Cholesky decomposition techniques.We state and prove the existence of first order shape derivative dE for interface Maxwell's equations(4.2)?(4.6)in following lemma.Lemma 5 Assume the deterministic perturbed interface ?? ? C2,1(?.R3),and ?0 is sufficiently small to ensure that the interface ?? is not degenerate and lies still inside the doma in D.Then the shape derivative dE exists and yieldswith interface conditions(?)on ? and boundary condition (?)With the shape Taylor expansion(4.49),noting the linearity of expectation operator E and the zero mean filed assumption(4.12),we have following lemma for approximating EE:Lemma 6 Under the assumptions of Theorem 4.3,then the following approximation holds EE(?)By using the relation(4.13)and simple calculations,we can derive following identity VardE(?)Once with the two-point correlation matrix CordE(x,y)at hand,using the identity(4.51),we can approximate the variance VarE(x)by the following Theorem.Lemma 7 Under the assumptions of Theorem 4.3,then the following estimate holds for the variance VarE(x)of random solution of(4.14) asWe also present the Nedelec element method for solving the deterministic equation of Eh in(4.54),which gives a second order approximation of the expectation.An efficient computation for the variance of the random interface Maxwell's equations is also proposed,which is approximated of third order by computing the CordE based on a low-rank ap-proximation via the pivoted Cholesky decomposition.The numerical simulations verify the efficiency of our algorithms.In the conclusion,this thesis mainly study on the numerical methods for the ran-dom Brinkman problem,the random interface Helmholtz problem and random interface interface Maxwell problem.Theoretically,we present efficient numerical algorithms and prove prove their convergence.Numerically,we verify the proposed methods are correct,efficient and practical.