Conditioning of implicit Runge–Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

The conditioning of implicit Runge–Kutta (RK) integration for linear finite element approximation diffusion equations on general anisotropic meshes is investigated. Bounds are established the condition number resulting system with and without diagonal preconditioning Euler (the simplest RK method) methods. Two solution strategies considered from integration: simultaneous where solved as a whole successive which follows commonly used implementation methods to first transform into smaller systems using Jordan normal form matrix then solve them successively. For in case positive semidefinite symmetric part coefficient it shown that an method behaves like method. If smallest eigenvalue negative strategy used, upper bound time step given so definite. obtained bounds have explicit geometric interpretations take interplay between mesh geometry full consideration. They show there three mesh-dependent factors can affect conditioning: elements, nonuniformity measured Euclidean metric, respect inverse matrix. also reveal matrix, mass or lumped effectively eliminate effects metric. Illustrative numerical examples given.