Stable Finite Elements for Problems with Wild Coefficients

We consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain. A piecewise linear finite element method is used. It is shown that the standard method for solving the resulting linear equations in finite-precision arithmetic can give an arbitrarily inaccurate answer because of ill-conditioning in the stiffness matrix. A new method for stably solving the linear equations is proposed. This method is based on a “mixed formulation” and gives a numerically accurate answer independent of the variation in the coefficients. The error bound for the solution of the linear system by the new method is shown to depend on the angles of the triangulation.