Since their introduction over fifty years ago, alternating direction implicit (ADI) methods have been employed to solve a variety of time-dependent multidimensional problems. Their primary attraction is that they reduce such a problem to independent systems of one-dimensional problems.
In this paper, we describe an ADI method for the solution of a class of two-component nonlinear reaction-diffusion problems on evolving domains. In this method, orthogonal spline collocation (OSC) is used for the spatial discretization, and the time-stepping is done with an algebraically linear ADI method based on an extrapolated Crank-Nicolson OSC method. Numerical results are provided to demonstrate the expected global rates of convergence of the method.
In this talk, we shall present preliminary results based on the Arnold-Falk-Winther mixed finite element for the linear elasticity. The method introduces the stress tensor, the displacement field, and the anti-symmetric part of the strain tensor as independent variables. The formulation is based on the variant of the Hellinger-Reissner variational principle which weakly imposes the symmetry conditions on the stresses. The equations will be reformulated in terms of differential forms using the Finite Element Exterior Calculus. The simplest low order compatible discretization will be chosen. Numerical results will be shown on various planar linear elasticity problems. Special attention will be paid to the incompressible limit as the Poisson’s ratio tends to $1/2$.
The Dirichlet biharmonic equation occurs in many areas of science and engineering, including fluid mechanics, elasticity, material science, etc. It is a fourth order partial differential equation (PDE) which means that the numerical solution of this equation is far more difficult than second order PDEs such as the Poisson equation. We shall use the preconditioned conjugate method to solve the finite element problem in a complexity proportional to the number of unknowns, provided an optimal Poisson solver is available. The crucial step is to find a preconditioner based on the Poincare-Steklov operator (or Dirichlet to Neumann map) for a pseudodifferential operator. This method works for smooth domains in any number of space dimensions. It builds upon the fundamental work by Glowinski and Pironneau. Extensions to self-adjoint elliptic and parabolic fourth order PDEs will also be given.
In this talk we describe recent progress on numerical strategies for computing the spectrum of the Laplacian in situations where Dirichlet data is prescribed on part of the boundary, and Neumann data is prescribed on the rest. Such problems are known to possess poor regularity, especially if the Dirichlet-Neumann junction occurs at angles of $\geq \pi/2$. We present two numerical strategies- one based on boundary integral solvers, and the other on the use of conforming and non-conforming finite elements - in this context.