本論文是解橢圓介面問題(elliptic interface problem). 為了有更準確的結果，我們用多尺度有限元素法(multiscale finite element method）來求解。多尺度有限元素法主要由兩個部分組成，分別是局部(local) 多尺度基底函數和這些基底函數所組成的總體粗網格(global coarse formulation). 由於多尺度基底函數的準確與否會直接影響數值結果，因此設定多尺度基底函數邊界會是重要環節。本研究旨是為多尺度有限元素法提出一個好的離散方式，求解橢圓介面問題的數值PDE. 此方式透過迭代過程對局部粗網格邊界值更新得以讓粗網格(coarse-grid) 與細網隔(fine-grid)間資訊做交換，得到原始PDE的解做為設定下一次迭代局部粗網格所須邊界條件的資訊，避免花費昂貴的計算成本直接解細網格的資訊。另外，迭代法的操作對於數值誤差有消除的效果，因此每次計算得到的細網格數值不須要求高精確度的計算結果，透過迭代的過程得以修正誤差。
然而，為了得到更準確的細網格數值解做為局部粗網格邊界條件的設定，在每次迭代中所得到細網格的數值解會透過數值迭代法做幾步smoothing的迭代運算以削去局部粗網格邊界點上震盪的高頻誤差(high frequency error)，如此也有利加速得到數值結果的收斂。針對被介面曲線通過的局部細網格而言，為了有更吻合的數值估算結果，將採取immersed finite element基底的方式做計算。
在我們數值結果中，i-ApMsFEM 這方式能夠有效消除介面上和粗網格邊界上的誤差。此外L2-norm可達二次收斂和H1-norm為一次收斂的結果; 誤差值也能和係數比例(contrast ratios)呈獨立關係，這些結果都符合理論敘述。

This thesis is to develop an accurate and effective numerical scheme for solving elliptic interface problem based on a framework of the multiscale finite element method. The two major ingredients of multiscale finite element method are the construction of multiscale basis
functions on coarse elements and the global coarse formulation using these basis functions. The multiscale basis functions are defined to satisfy the original governing partial differential equation restricted in a coarse element with some proper boundary condition. The choice of the boundary condition for multiscale basis functions is important since it affects the accuracy of the global solution significantly. In this work, we propose an adaptive multiscale finite element method through an iterative process to improve the boundary settings for the multiscale basis functions, namely i-ApMsFEM. To demonstrate the capability of the proposed i-ApMsFEM to a variety of high contrast elliptic interface problems, we consider a series of testing problems. An immersed finite element method is used for constructing numerically multiscale basis and for formulating the coarse-grid problem. Our numerical
results show that i-ApMsFEM effectively eliminate the errors along the interface curve and the boundaries of coarse elements to produce highly accurate solutions. In addition, the second-order rate convergence in the L2-norm (or first-order in the H1-norm) is achieved which is independent of the contrast ratios.

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