Combined Multiscale Finite Element Methods For Multiscale Problems In Domains With Rough Boundaries

Many problems in nature and industry applications are described by partial differ-ential equations in domains with rough boundaries.Typical examples include fluid in rough domains,wetting phenomena on plant leaves with tiny roughness,wave scatter-ing on objects with oscillating surfaces,chemically reacting flow on a surface with a given microstructure and among many others.In this dissertation,we propose a com-bined finite element method of elliptic problems in domains with rough boundaries and an oversampling combined multiscale finite element method of high oscillating coeffi-cient elliptic problems in domains with rough boundaries,and apply the combined finite element method into parabolic problems in domains with rough boundaries,and study how to use localized orthogonal decomposition(LOD)technique to construct the com-bined multiscale finite element method for non-periodic coefficient elliptic problems in domains with rough boundaries.In Chapter 2,we study the combined finite element method of elliptic problems in domains with rough boundaries.The basic idea of the proposed method is to use a fine mesh with h in the vicinity of oscillating boundary and a coarse mesh with size H?h for other portions of the domain to reduce some unnecessary computational effort.The transmission conditions across the fine-coarse mesh interface is treated by penalty technique.The key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form,which avoiding the affection of the ratio H/h in the error estimate.We prove a quasi-optimal convergence in terms of elements since there is generally no whole H2 regularity in the domains with rough boundaries.Numerical results are provided for elliptic equations with non-oscillating or oscillating boundary to illustrate the theoretical results.In Chapter 3,for parabolic problems in domains with rough boundaries,we use the numerical method of the combined finite element method discretization in space and Euler-backward scheme in time.For symmetric(? = 1)and non-symmetric(? ?-1)bilinear form,we derive L2 error estimate of elliptic projection operator by dual argument,which depends on the regularities of two auxiliary problems(elliptic problem and one similar to interface problem).According to the idea of[99],we prove the convergence of the solution to parabolic problem in domain with rough boundaries in energy norm.Since there are no explicit convergence rates about space mesh size h,H and time step At in our error estimates,we study their convergence orders by numerical experiments.Numerical results suggest that the convergence rates of the introduced combined finite element method are not influenced by the oscillating of rough boundary,and convergence orders are quasi-optimal.In Chapter 4,we propose an oversampling multiscale combined finite element method to solve elliptic problems with oscillating coefficients in domains with rough boundaries.The basic idea of this method is to use the standard finite element method on a fine mesh with h in the vicinity of oscillating boundary,the oversampling multi-scale finite element method on a coarse mesh with size H>>h for other portions,and penalty technique on the interface.under the hypothesis of the periodicity of diffusion coefficient,we prove the stability and convergence of numerical scheme,and carry out error estimates.At last,we illustrate the efficiency of numerical method by numerical experiments.For non-periodic coefficient,there exist some difficulties in theoretical analysis,but numerical scheme itself is suitable.For random coefficient problems,we supply some numerical experiments to verify its effectiveness.In Chapter 5,we study how to make use of the localized orthogonal decomposition technique to construct the combined multiscale finite element method for solving mul-tiscale elliptic problems with non-periodic coefficients in domains with rough bound-aries.The idea of localized orthogonal decomposition is to decompose the solution s-pace into two subspaces.The two subspaces are orthogonal about the inner product that the bilinear form defines.The useful space of which has low dimension and includes partial microscopic information.However in order to obtain these information,we of-ten solve many subproblems whose computational effort accounts to that of the original problem,which isn't possible in reality.Hence we need to localize these subproblems.In this chapter,we first recall the localized orthogonal decomposition technique and lo-calized orthogonal decomposition multiscale finite element method,and introduce the properties of basis functions in finite element space,show the feature of basis functions by diagrams,then use localized orthogonal decomposition to construct a new finite ele-ment space,finally propose a numerical scheme of combined finite element method for solving multiscale elliptic problem with non-periodic coefficients.