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Author: Lang, Jens × DDC: 510 × Subject: Finite element method × WGL Institute: WIAS ×

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    Stability of explicit Runge-Kutta methods for high order finite element approximation of linear parabolic equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Huang, Weizhang; Kamenski, Lennard; Lang, Jens
    We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.
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    Stability of explicit Runge-Kutta methods for finite element approximation of linear parabolic equations on anisotropic meshes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Huang, Weizhang; Kamenski, Lennard; Lang, Jens
    We study the stability of explicit Runge-Kutta integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of 2(d + 1), where d is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.