A moving grid finite element method applied to a model biological pattern generator
Journal of Computational Physics
Approaches for generating moving adaptive meshes: location versus velocity
Applied Numerical Mathematics - Special issue: 2nd international workshop on numerical linear algebra, numerical methods for partial differential equations and optimization
Stabilized Moving Finite Elements for Convection Dominated Problems
Journal of Scientific Computing
Journal of Scientific Computing
Special issue on the method of lines: dedicated to Keith Miller
Journal of Computational and Applied Mathematics - Special issue on the method of lines: Dedicated to Keith Miller
Nonlinear Krylov and moving nodes in the method of lines
Journal of Computational and Applied Mathematics - Special issue on the method of lines: Dedicated to Keith Miller
A new adaptive mesh refinement strategy for numerically solving evolutionary PDE's
Journal of Computational and Applied Mathematics
A moving mesh method with variable mesh relaxation time
Applied Numerical Mathematics
Editorial: Special Issue on the Method of Lines: Dedicated to Keith Miller
Journal of Computational and Applied Mathematics - Special issue on the method of lines: Dedicated to Keith Miller
Nonlinear Krylov and moving nodes in the method of lines
Journal of Computational and Applied Mathematics - Special issue on the method of lines: Dedicated to Keith Miller
Anderson Acceleration for Fixed-Point Iterations
SIAM Journal on Numerical Analysis
A bidirectional coupling procedure applied to multiscale respiratory modeling
Journal of Computational Physics
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This paper reports on the design of a robust and versatile gradient-weighted moving finite element (GWMFE) code in one dimension and its application to a variety of difficult PDEs and PDE systems. A companion paper, part II, will do the same for the two-dimensional (2D) case. These moving node methods are especially suited to problems which develop sharp moving fronts, especially problems where one needs to resolve the fine-scale structure of the fronts. Brief explanations are given of the variational interpretation of GWMFE, the geometrical-mechanical interpretation, simplified regularization terms, and the treatment of PDE systems. There are many possible pitfalls in the design of GWMFE codes; section 5 discusses special features of the implicit one-dimensional (1D) and 2D codes which contribute greatly to their robustness and efficiency. Section 6 uses a few simple examples to illustrate the workings of the method, some difficulties, and reasons for the standard choices of the internodal viscosity regularization coefficient. Section 7 reports numerical trials on several more difficult PDE systems. Section 8 discusses the failure of the method on certain steady-state convection problems. Section 9 describes a simple nonlinear "Krylov subspace" accelerator for Newton's method, a routine which greatly decreases the number of Jacobian evaluations required for our stiff ODE solver.