Optimal finite element mesh for elliptic equation of divergence form
The authors discuss the generation of triangular meshes for elliptic partial differential equations that result in stiffness matrices with minimal condition numbers. The condition number \(\mathcal K\) of the stiffness matrix for the Dirichlet problem is estimated in terms of the elemental quantities \({\mathcal K}_e\), defined as the ratio of the largest eigenvalue of the elemental stiffness matrix divided by the smallest eigenvalue of the elemental mass matrix. Only linear finite elements on triangles are considered. A direct calculation for the Poisson equation shows that the quantity \({\mathcal K}_e\) is minimized for equilateral triangles. For right triangles, it is minimized for isosceles right triangles. For a general coercive elliptic equation with coefficient matrix \(A\), the authors show that the metric induced by the matrix \(A^{-1/2}\) should be used. When the coefficient matrix varies with position, optimality can be approximated using the value of \(A\) at each element's centroid. These results give rise to a mesh generation algorithm based on generating equal-legged triangles in the \(A^{-1/2}\) metric. Although no proof is presented that this mesh strategy results in the minimal condition number \(\mathcal K\), a direct calculation for the case of a uniform isosceles right triangular mesh for the Poisson equation shows that \(\mathcal K\) agrees with each \({\mathcal K}_e\), supporting the likelihood that the strategy does minimize \(\mathcal K\). In addition, several numerical examples are presented.
scientific article; zbMATH DE number 2137729
scientific article