本論文是針對 multiscale finite element method with adaptive bubble enrichment method (MsFEM\_bub)提出一種改進方法，期望使用更少的計算成本，達到與原數值方法相似的精確度，並藉由模擬對流-擴散-反應問題(convection-diffusion-reaction problem)，呈現並比較數值結果。
我們發現那些真正需要更新的粗網格元素，無論是多尺度基底函數 (multiscale basis functions) 或是氣泡函數 (bubble function)，通常都在數值結果產生劇烈變化的地方，而那些在平滑的地方，或是計算前後解改變幅度很小的粗網格元素，是不需要更新的，可是 MsFEM\_bub 會將全部的粗網格元素都進行更新，這造成計算成本的浪費。所以根據這樣的想法，我們提出 multiscale finite element with local adaptive bubble function enhancement，只針對需要更新的粗網格元素進行計算，並對於其他不需要更新的粗網格元素不進行計算。實驗後發現透過這樣的方法，我們可以在減少計算成本的情況下，達到與 MsFEM\_bub 相近的數值結果。

In this thesis, we propose an improved method for multiscale finite element method with adaptive bubble enrichment method (MsFEM\_bub). We expected to use less computation cost to get the similar accuracy as the original numerical method. We would show and compare the numerical solutions by simulating the convection-diffusion-reaction problems. \\
We found that the coarse-grid elements that really need to update, no matter multiscale basis functions or bubble functions, are usually at the sharp place of the numerical solutions, and those coarse-grid elements which are at the smooth place, or a small change of the solution before and after update, do not need to update, but MsFEM\_bub will update all the coarse-grid elements, which would cause the waste of computation costs. So according to this idea, we propose multiscale finite element with local adaptive bubble function enhancement, only update the coarse-grid elements for those really need, and do not update for the others not need. After the experiment, we found that through this method, we can reduce the computation costs and get the similar numerical solutions as MsFEM\_bub.

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