【摘要】：弱有限元(Weak Galerkin,簡稱WG)方法首先是由王軍平和葉秀等人提出利用弱函數(shù)和弱梯度來求解二階橢圓問題.弱函數(shù)空間的選取依賴于定義在網(wǎng)格單元的內(nèi)部和邊界上的多項(xiàng)式空間,這使得弱有限元方法在許多應(yīng)用上更加的靈活可靠.一般來說,間斷有限元(Discontinuous Galerkin)方法定義的跳躍來自分片單元內(nèi)部函數(shù)在單元邊界上函數(shù)值;弱有限元方法傳統(tǒng)做法是在單元邊界上定義單值函數(shù),與內(nèi)部單元函數(shù)利用弱函數(shù)定義聯(lián)系.本文與之區(qū)別在于,我們在單元邊界上定義雙值函數(shù),于是在同一個單元邊界上就自然產(chǎn)生兩個弱函數(shù)的差,我們稱之為弱跳躍.基于內(nèi)罰間斷Galerkin有限元的思想,我們對單元邊界上的弱跳躍加罰項(xiàng),就形成本文介紹的超罰弱有限元方法.本文主要以具有光滑解的二階橢圓問題為例討論超罰弱有限元方法.文中首先給出了二階橢圓問題的數(shù)值格式,嚴(yán)格證明了在H1-范數(shù)和L2-范數(shù)意義下基于弱函數(shù)空間(Pk,Pk,RTk)(k≥0)的先驗(yàn)誤差估計,并且給出相關(guān)的數(shù)值實(shí)驗(yàn)來驗(yàn)證理論結(jié)果.本文還給出另外一種同時帶有超罰項(xiàng)和穩(wěn)定項(xiàng)的WG方法,它基于弱函數(shù)空間(Pk,Pk,[Pk-1]2)(k≥1)或者(Pk,Pk-1,[Pk-1]2)(k≥1).該算法格式中網(wǎng)格剖分單元不再局限于單純形,而是可以擴(kuò)展到一般多邊形或多面體,并且理論和實(shí)驗(yàn)都表明,通過選取適當(dāng)?shù)牧P參數(shù)同樣可以達(dá)到最優(yōu)收斂階.從算法的格式和數(shù)值實(shí)現(xiàn)來看,它是弱有限元方法的一種自然拓廣,具有逼近函數(shù)簡單,網(wǎng)格生成靈活,并且其數(shù)值格式絕對穩(wěn)定,單元剛度矩陣可以獨(dú)立實(shí)現(xiàn),便于并行計算等優(yōu)點(diǎn).值得一提的是,本文雖然只給出了二階橢圓問題的超罰弱有限元格式,但是可以將這個方法應(yīng)用到其它常見的偏微分方程上,如橢圓界面問題,Stokes方程,div-curl系統(tǒng)等.
[Abstract]:The weak finite element (Weak Galerkin, (WG) method is firstly proposed by Wang Junping and Ye Xiu to solve the second order elliptic problem by using weak function and weak gradient. The selection of the weak function space depends on the polynomial space defined on the interior and boundary of the grid element, which makes the weak finite element method more flexible and reliable in many applications. In general, the jump defined by the discontinuous finite element (Discontinuous Galerkin) method comes from the function value of the internal function of the piecewise element on the boundary of the element. The traditional method of weak finite element method is to define the single-valued function on the boundary of the element, which is related to the definition of the internal element function by using the weak function. The difference between this paper and the other is that we define the two-valued function on the element boundary, so the difference between the two weak functions on the same cell boundary is naturally generated, which is called weak jump. Based on the idea of discontinuous Galerkin finite element with internal penalty, we add penalty term to the weak jump on the boundary of the element, and form the super-penalty weak finite element method introduced in this paper. In this paper, the second order elliptic problem with smooth solution is taken as an example to discuss the superpenalty weak finite element method. In this paper, the numerical scheme of the second order elliptic problem is given, and a priori error estimate based on weak function space (Pk,RTk) (k 鈮，

本文編號：2312392