邊界層和內部層是空間定義域中使得統御方程式解的微分值變化快速的狹長區域，此種具有邊界層或內部層的解在眾多工程與科學的應用問題中是極常見的特徵，然而傳統處理此類問題的數值方法常常缺乏穩定性或精確度。在這兩年期計畫中，我們將致力於開發高效率數值方法求取邊界層和內部層問題的精確與穩定近似解，我們將考慮包含多重尺度有限元素法和裁製有限點法等兩大類解法。二階對流佔優的對流-擴散偏微分方程式的解可能呈現邊界層和內部層現象應該是眾所皆知的事實，另外此類方程亦被視為一種研究較高雷諾數的不可壓縮那維爾-史托克方程組的簡易捷徑，因此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高雷諾數不可壓縮那維爾-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我們所開發的方法具有高精確度與穩定性。 Boundary and interior layers are narrow regions in the spatial domain, where some derivative of solution of the governing equations rapidly changes. Such layer structures in the solution are a familiar feature of certain classes of applications in engineering and science. However, most of the conventional numerical methods for layer problems are lacking in either stability or accuracy. In this two-year project, we will devote to the development of efficient numerical methods for solving boundary and interior layer problems. The numerical methods under consideration include the multiscale finite element methods and the tailored finite point methods. It is well known that the convection-dominated convection-diffusion equation may exhibit boundary and interior layers in its solution, and it also serves as a vehicle to study the more advanced incompressible Navier–Stokes equations with a high Reynolds number. Therefore, as a model problem, we will start with a scalar second-order convection-dominated convection-diffusion problem, and then extend the investigations to the incompressible Navier-Stokes problem and the incompressible magnetohydrodynamic problem. Numerical experiments will be performed to demonstrate the accuracy and stability of the efficient numerical methods. 研究期間：10008 ~ 10107
