二维泊松方程的虚拟有限元方法数值分析.pdf

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Introduction

The continuous problem and discretization

The construction of VEM and error analysis

Numerical case

Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case The Virtual Element Method on the two-dimensional Laplace equation June 27, 2020

Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case content Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case

Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case Introduction The virtual element method is a generalisation of the standard conforming ﬁnite element method for the approximation of solutions to partial diﬀerential equations. The method is designed in such a way as to enable the construction of high order approximation spaces which may include an arbitrary degree of global regularity on meshes consisting of very general polygonal (or polyhedral) elements. In this report, We preserve the generality of the shape of the elements in the decomposition of the computational domain, and the generality in the degree k of accuracy that we require to the method.Finally, We give the numerical example.

Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case Throughout the paper, we will follow the usual notation for Sobolev spaces and norms.In particular,for an open bounded do- main D,we will use | · |s,D and · s,D to denote seminorm and norm,respectively,in the Sobolev space H s (D),while (·,·)0,D will de- note the L2(D) inner product.Often the subscript will be omit- ted when D is the computational domain Ω.For k a non-negative integer,Pk (D) will denote the space of polynomials of degree ≤ k on D.Conventionally, P−1(D) = 0. Moreover, P D k will denote the usual L2(D)-orthogonal projection onto Pk (D). Finally, C will be a generic constant independent of the decomposition that could change from one occurrence to the other.

Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case The continuous problem and discretization We consider the problem − u = f in Ω, u = 0 on Γ = ∂Ω, (2.1) where Ω ⊂ R2 is a polygonal domain and f ∈ L2(Ω). The variational formulation reads:ﬁnd u ∈ V := H 1 0 (Ω) such that a(u, v ) = (f , v ) ∀v ∈ V , (2.2) with(·,·)=scalar product in L2,a(u, v ) = (u,v ),|v|2 1 = a(v , v ).

Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case It is clear that seminorm | · |1 is a norm on H 1 0 (Ω),because if we set |v|1 = 0,then with Poincar´e inequality, we have v1 ≤ C|v|1 = 0,so we have v ≡ 0 on Ω.With Cauchy Schwartz inequality,it is also well known that a(v , v ) ≥ |v|2 1 and a(u, v ) ≤ |u|1|v|1,∀u, v ∈ V (2.3) So problem (2.2) has a unique solution.

Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case The discrete problem Let {Th}h be a sequence of decompositions of Ω into elements K ,and let εh be the set of edges e of Th.As usual,h will also denote the maximum of the diameters of the elements in Th. Next,we give some assumption: A0.1. For every h, the decomposition Th is made of a ﬁnite number of simple polygons (meaning open simply connected sets whose boundary is a non-intersecting line made of a ﬁnite number of straight line segments).

Introduction The continuous problem and discretization The construction of VEM and error analysis Numerical case The bilinear form a(·,·) and the norm | · |1 can obviously be split as  a(u, v ) = |v|1 = ( to the space H 1(Th) := K∈{Th}h K∈{Th}h H 1-seminorm: aK (u, v ) ∀u, v ∈ V , |v|2 1,K )1/2 ∀v ∈ V . (2.4) Since in what follows we shall also deal with functions belonging H 1(K ), we need to deﬁne a broken K∈Th |v|h,1 := ( K∈TH |∇v|2 0,K )1/2 (2.5) Note that, for discontinuous functions, this is really a seminorm and not a norm: for instance, |Ch|h,1 ≡ 0 for every piecewise constant function ch.

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二维泊松方程的虚拟有限元方法数值分析.pdf

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