我們利用曲線座標建構一個新的差分方法來求有介面橢圓方程的數值解。所謂有介面橢圓方程是指其擴散係數、方程解本身或其流量有不連續面。我們的數值方法可以很容易地經由對四周的點值進行差分及平均而產生。同時，這個方法是對稱且正定的，所以可以利用PCG或多重網格的迭代法來進行線性系統的反解。

這個方法是「單調的」，也就是說它滿足極值原理，所以我們可以証明它是二階收斂的。我們也提出了很多的例子以及數值結果，顯示解和流量的誤差都有二階的收斂。

我們的方法自動地考慮到介面上的不連續條件，不需要額外的計算。這是此方法最大的好處，因為這樣可以保持數值解的不連續性質，在介面上不會有光滑化的失真現象。這個新的方法可以很直接地推廣到多相流的問題上，並且可以結合區域分割的方法來處理複雜形狀的區域和介面。

In this dissertation, we use body-fitting curvilinear coordinates to construct a finite difference scheme, called the Monotone Jump Condition Capturing Scheme (MJCCS), for the elliptic interface problems. The variable coefficients, the solution itself and its normal flux may be discontinuous across the interface. The entries of the coefficient matrix are easily generated via centered difference and average on neighboring mesh points. The resultant matrix of MJCCS is symmetric and positive definite. Most powerful linear solvers such as PCG and multigrid can be applied to invert the matrix.

The scheme preserves monotonicity on mild distorted grids. It is proved to achieve second-order accuracy in the sup-norm of global errors. Extensive numerical experiments are performed to demonstrate the second-order convergence rate in the numerical solution and the reconstructed flux.

Our scheme automatically captures the jump conditions in the difference formulation without extra enforcement. This is the main advantage of the new scheme. The approximation obtained by our scheme resolves the discontinuities at the interface without numerical smearing at all. Besides, one can apply the scheme to treat more general discontinuities such as multi-phase flows in a straightforward manner. We also extend the idea incorporating with the nonoverlapping domains decomposition method to tackle the domains and interfaces with complicated geometry.

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