We present a simple and direct method for solving the Beltrami equation {partial deriv}f/{partial deriv}z = μ{partial deriv}f/{partial deriv}z for a quasiconformal self-mapping of planar disk without approximating singular integrals or infinite series. Given a triangulation of the disk, it is a simple matter to discretize the Beltrami equation as a sistem of linear equations. However, natural attempts to require the boundary to be a circle (so that the disk is mapped to a disk) normally produce nonlinear conditions which would require iterative methods for the solution. We show how to eliminate all nonlinear conditions, and prove that the Least-Squares solution to the linear system is a good approximation for the solution of the Beltrami equation in the sense that as the triangular mesh is refined appropriately, the discrete solution approaches the true μ-conformal mapping.
展开▼
机译：我们介绍了一种简单而直接的方法来解决Beltrami等式{部分德国} f / {partive deriv} f / {partial deriv} f / {eartial deriv} f / {部分deriv} f / {部分deriv} z，用于平面盘的quasiconformal自映射而不近似奇异积分或无限系列。鉴于磁盘的三角测量，这是一个简单的事情，以将Beltrami等式离散作为线性方程的sistem。然而，自然尝试要求边界是圆圈（使磁盘映射到盘）通常会产生非线性条件，这需要迭代方法进行解决方案。我们展示了如何消除所有非线性条件，并证明线性系统的最小二乘解是对Beltrami方程的解决方案的良好近似，因为当三角网格被适当地精制时，离散解决方案接近真实μ-保形映射。
展开▼
