在這兩年期計畫中，我們將致力於研究多重尺度有限元素法求解對流佔優問題，其基本思想在結合最小平方有限元素法與無殘餘泡狀函數法兩者優勢。在引入一個或多個新的物理量如速度、旋度或應力後，原始二階偏微分方程組將可重寫為一個一階方程組。針對此一階方程組，在粗網格上我們引用最小平方有限元素法求其數值解，其中所使用的片狀線性有限元將以無殘餘泡狀函數補強，此泡狀函數的選取要求補強後的有限元在每一個小單元內部滿足原始二階偏微分方程，但在單元邊界上為零值(即所謂無殘餘泡狀函數)。其中的最小平方能量泛函定義為殘餘方程組在適當泛函空間的L2模之平方和，而最小平方有限元素解就定義為該最小平方能量泛函在強化型有限元素空間的最小值函數。我們將先採用一個二階對流佔優的對流擴散偏微分方程式為模型問題，進而將此研究方法推廣至不可壓縮那維爾-史托克方程組和不可壓縮磁流體方程組問題上，同時亦將進行數值實驗模擬來實證此類多重尺度最小平方有限元素法的精確度與穩定性。 ; In this two-year project, we devote to the study of multiscale finite element methods for solving the convection-dominated problems. The basic idea is to combine the advantages of least-squares finite element method with the residual-free bubble method. Introducing one or more additional physically unknown variables such as velocity, vorticity or stress, one can reformulate the original second-order partial differential equations into a first-order system. For such a first-order system, we apply the least-squares finite element method on relatively coarse meshes to solve the problem by enriching piecewise linear polynomials with bubble functions. The bubble function is chosen so that the computed solution satisfies the original second-order differential equation in the interior of each element and vanishes on the boundary of each element. The associated least-squares energy functional is defined to be the sum of squared L2 norms of the residual equations over an appropriate product space, and the least-squares finite element solution is defined to be the minimizer of the associated functional over the enriched finite element space. As a model problem, we will start with a scalar second-order convection-dominated convection-diffusion problem, and then extend the approach to the incompressible Navier-Stokes problem and the incompressible magneto-hydrodynamic problem. Numerical experiments will be performed to demonstrate the accuracy and stability of the multiscale least-squares finite element approach. ; 研究期間 9708 ~ 9807
