Talal Rahman

Fast MATLAB assembly of FEM matrices in 2D and 3D: Nodal elements

We propose an effective and flexible way to assemble finite element stiffness and mass matrices in MATLAB. The major loops in the code have been vectorized using the so called array operation in MATLAB, and no low level languages like the C or Fortran has been used for the purpose. The implementation is based on having the vectorization part separated, in other words hidden, from the original code thereby preserving its original structure, and its flexibility as a finite element code. The code is fast and scalable with respect to time.

Additive Schwarz Methods for Elliptic Mortar Finite Element Problems

Summary.Two variants of the additive Schwarz method for solving linear systems arising from the mortar finite element discretization on nonmatching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed. The methods are defined on subdomains without overlap, and they use special coarse spaces, resulting in algorithms that are well suited for parallel computation. The condition number estimate for the preconditioned system in each method is proportional to the ratio H/h, where H and h are the mesh sizes, and it is independent of discontinuous jumps of the coefficients. For one of the methods presented the choice of the mortar (nonmortar) side is independent of the coefficients.

Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients

In this paper, we propose two variants of the additive Schwarz method for the approximation of second order elliptic boundary value problems with discontinuous coefficients, on nonmatching grids using the lowest order Crouzeix-Raviart element for the discretization in each subdomain. The overall discretization is based on the mortar technique for coupling nonmatching grids. The convergence behavior of the proposed methods is similar to that of their closely related methods for conforming elements. The condition number bound for the preconditioned systems is independent of the jumps of the coefficient, and depend linearly on the ratio between the subdomain size and the mesh size. The performance of the methods is illustrated by some numerical results.

A TV-stokes denoising algorithm

In this paper, we propose a two-step algorithm for denoising digital images with additive noise. Observing that the isophote directions of an image correspond to an incompressible velocity field, we impose the constraint of zero divergence on the tangential field. Combined with an energy minimization problem corresponding to the smoothing of tangential vectors, this constraint gives rise to a nonlinear Stokes equation where the nonlinearity is in the viscosity function. Once the isophote directions are found, an image is reconstructed that fits those directions by solving another nonlinear partial differential equation. In both steps, we use finite difference schemes to solve. We present several numerical examples to show the effectiveness of our approach.

A Modified TV-Stokes Model for Image Processing

We introduce and investigate the modified total variation (TV)-Stokes model for two classical image processing tasks, i.e., image restoration and image inpainting. The modified TV-Stokes model is a two-step model based on a TV minimization in each step and the use of geometric information of the image. In the first step, a smoothed and divergence-free tangential field of the given image is recovered, and in the second step, the image is reconstructed from the corresponding normals. The existence and the uniqueness of the solution to the minimization problems are established for both steps of the model. Numerical examples and comparisons are presented to illustrate the effectiveness of the model.

The Crouzeix-Raviart FE on Nonmatching Grids with an Approximate Mortar Condition

A new approximate mortar condition is proposed for the lowest order Crouzeix-Raviart finite element on nonmatching grids, which uses only the nodal values on the interface for the calculation of the mortar projection. This approach allows for improved and more flexible algorithms compared to those for the standard mortar condition where nodal values in the interior of a sub-domain, those closest to a mortar side of the subdomain, are also required in the calculation.

An additive average Schwarz method for the plate bending problem

Abstract The original additive average Schwarz method and its variants were introduced for solving second order elliptic boundary value problems. In this paper, we extend the original idea to fourth order problems by designing a variant of the additive average Schwarz method for the plate bending problem using the nonconforming Morley finite element for the discretization. Like the original average method this method uses only nonoverlapping subdomains, and its coarse space does not need any explicit coarse mesh. An analysis of the method, and some numerical experiments to validate the theory are presented.

Problems of Stationary Flow of Electrorheological Fluids in a Cylindrical Coordinate System

We consider the general problem on stationary flow of the electrorheological fluid with the constitutive equation developed in [R. H. W. Hoppe and W. G. Litvinov, Comm. Pure. Appl. Anal., 3 (2004), pp. 809--848] in the cylindrical coordinate system. The problem is studied under mixed boundary conditions wherein velocities are specified on one part of the boundary and surface forces are given on the other part. The existence of a solution to this problem and the convergence of Galerkin approximations are established. Then, we consider the occasion where the flow is axially symmetric and study a problem on an electrorheological clutch. This problem is solved numerically, and the results of calculations of the electric field and velocities are presented.

A Dual Formulation of the TV-Stokes Algorithm for Image Denoising

We propose a fast algorithm for image denoising, which is based on a dual formulation of a recent denoising model involving the total variation minimization of the tangential vector field under the incompressibility condition stating that the tangential vector field should be divergence free. The model turns noisy images into smooth and visually pleasant ones and preserves the edges quite well. While the original TV-Stokes algorithm, based on the primal formulation, is extremely slow, our new dual algorithm drastically improves the computational speed and possesses the same quality of denoising. Numerical experiments are provided to demonstrate practical efficiency of our algorithm.

Additive average Schwarz method for a Crouzeix---Raviart finite volume element discretization of elliptic problems with heterogeneous coefficients

In this paper we introduce an additive Schwarz method for a Crouzeix–Raviart finite volume element discretization of a second order elliptic problem with discontinuous coefficients, where the discontinuities are both inside the subdomains and across and along the subdomain boundaries. We show that, depending on the distribution of the coefficient in the model problem, the parameters describing the generalized minimal residual method (GMRES) convergence rate of the proposed method depend linearly on the mesh parameters . Also, under certain restrictions on the distribution of the coefficient, the convergence of the GMRES method is independent of jumps in the coefficient.