Applied numerical linear algebra
Applied numerical linear algebra
Accurate projection methods for the incompressible Navier—Stokes equations
Journal of Computational Physics
Preserving monotonicity in anisotropic diffusion
Journal of Computational Physics
Journal of Computational Physics
An Asymptotic Preserving scheme for the Euler equations in a strong magnetic field
Journal of Computational Physics
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Solving elliptic PDEs in more than one dimension can be a computationally expensive task. For some applications characterized by a high degree of anisotropy in the coefficients of the elliptic operator, such that the term with the highest derivative in one direction is much larger than the terms in the remaining directions, the discretized elliptic operator often has a very large condition number - taking the solution even further out of reach using traditional methods. This paper will demonstrate a solution method for such ill-behaved problems. The high condition number of the D-dimensional discretized elliptic operator will be exploited to split the problem into a series of well-behaved one and (D-1)-dimensional elliptic problems. This solution technique can be used alone on sufficiently coarse grids, or in conjunction with standard iterative methods, such as Conjugate Gradient, to substantially reduce the number of iterations needed to solve the problem to a specified accuracy. The solution is formulated analytically for a generic anisotropic problem using arbitrary coordinates, hopefully bringing this method into the scope of a wide variety of applications.