Coefficient adaptive triangulation for strongly anisotropic problems

摘要

Second order elliptic partial differential equations arise in many important applications, including flow through porous media, heat conduction, the distribution of electrical or magnetic potential. The prototype is the Laplace problem, which in discrete form produces a coefficient matrix that is relatively easy to solve in a regular domain. However, the presence of anisotropy produces a matrix whose condition number is increased, making the resulting linear system more difficult to solve. In this work, we take the anisotropy into account in the discretization by mapping each anisotropic region into a ''stretched'' coordinate space in which the anisotropy is removed. The region is then uniformly triangulated, and the resulting triangulation mapped back to the original space. The effect is to generate long slender triangles that are oriented in the direction of ''preferred flow.'' Slender triangles are generally regarded as numerically undesirable since they tend to cause poor conditioning; however, our triangulation has the effect of producing effective isotropy, thus improving the condition number of the resulting coefficient matrix.