Preconditioners for elliptic problems via non-uniform meshes

In this paper, an approximate solution of non-linear parabolic partial differential equation is obtained for a non-uniform mesh. The scheme for partial differential equation subject to Neumann boundary is based on cubic B-spline collocation method. Modified cubic B-splines are proposed over non-uniform mesh to deal with the Dirichlet boundary conditions. This scheme produces a system of first order ordinary differential equations. This system is solved by Crank Nicholson method. The stability is also discussed using Von Neumann stability analysis. The accuracy and efficiency of the scheme is shown by numerical experiments. We have compared the approximate solutions with that in the literature.

Mesh stretch effects on convection in flow simulations

This paper is concerned with computational mesh of the Marker and Cell method. The comutational stability of the finite difference equations which is based on the MAC method, using non-uniform mesh, is examined with respect to trucation errors and energy conservance of the finite difference equation. And restrictions for accurate finite difference calculations, using Mesh Reynolds No. and Flow-Local Reynolds No., are proposed. Those are, (1) R_m<20 (2) R_m<√<R_l> R_m : Mesh Reynolds No. R_l : Flow-Local Reynolds No. These ristrictions are examined by numerical experiments of a few flow models, and proved to be useful for both uniform and non-uniform computational meshes of the finite difference equations.

A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes

Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted\(L^{2}\) -projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.

In the present work, we propose a two-phase fourth-order method for the approximate numerical solution for second-order non-linear two-point boundary value problems with Dirichlet boundary conditions. Our numerical approach is based on a finite difference and the solution of the problem at discrete points. Our method generates a system of equations, and the solution of the system of equations is considered an approximate solution to the problem. An essential analysis of the method is considered to ensure the performance of the method. A numerical experiment is carried out with model problems to test the performance in terms of efficiency and accuracy of the proposed method.

A computer code is described which solves Poisson's equation for the electric potential over a two dimensional cylindrical (r,z) nonuniform mesh which can contain internal electrodes. Poisson's equation is solved over a given region subject to a specified charge distribution with either Neumann or Dirichlet perimeter boundary conditions and with Dirichlet boundary conditions on internal surfaces. The static electric field is also computed over the region with special care given to normal electric field components at boundary surfaces.

3-D field solvers have become popular tools for parasitic capacitance extraction of full custom circuits, IPs, and packages. Traditional field solvers based on finite difference method/finite element method/floating random walk truncate the field on the outer boundary of the numerical region by applying Dirichlet or Neumann boundary conditions. However, to ensure high accuracy in the vicinity of circuit elements, a substantial gridded region between the circuit and the outer boundary is necessary in order to reduce distortion of the field caused by these truncation conditions. In this paper, we make a fundamental contribution to the application of numerical field solvers by proposing a class of absorbing boundary conditions, which when implemented, significantly reduce the distortion of the field at the numerical boundary, and consequently, throughout the numerical region. The absorbing boundary condition we propose will allow the field throughout the numerical region to behave as though there is no numerical boundary, accurately mimicking the fields in an actual circuit. As a result, the size of the numerical region can be significantly reduced, which in turn reduces the run time without sacrificing accuracy. A mathematical development of the proposed absorbing boundary condition is presented. It is shown that the error of the proposed ${n}$ th order absorbing boundary is ${O(1/r^{n+2})}$ , while the error of the traditional Neumann boundary is ${O(1/r^{2})}$ , where ${r}$ is the size of the numerical region. Experimental results for capacitance extraction with interconnects in multilayer dielectrics and silicon on insulator show the proposed methods improve the run time and accuracy of numerical solutions of Laplace’s equation over previous boundary conditions for uniform or nonuniform meshes.

A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid

Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems

A high accuracy hybrid method for two-dimensional Navier–Stokes equations

In this article, a family of high order accurate compact finite difference scheme for obtaining the approximate solution values of mildly nonlinear elliptic boundary value problems in three-space dimensions has been developed. The discretization formula is developed on a non-uniform meshes, which helps in resolving boundary and/or interior layers. The scheme involves 27 points single computational cell to achieve high order truncation errors. The proposed scheme has been applied to solve convection–diffusion equation, Helmholtz equation and nonlinear Poisson’s equation. A detailed convergence theory for the new compact scheme has been proposed using irreducible and monotone property of the iteration matrix. Numerical results show that the new compact scheme exhibit better performance in terms of \(l^\infty \)- and \(l^2\)-error of the exact and approximate solution values.

The finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains. Consider the following model problem: \[ − Δ u = f in Ω C , u = g on ∂ Ω , ∂ u ∂ r + 1 r u = o ( 1 r ) as r = | x | → ∞ , - \Delta u = f\quad {\text {in}}\;{\Omega ^C},\quad u = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial u}}{{\partial r}} + \frac {1}{r}u = o\left ( {\frac {1}{r}} \right )\quad {\text {as}}\;r = |x| \to \infty , \] where Ω C {\Omega ^C} is the complement in R 3 {R^3} (three-dimensional Euclidean space) of a bounded set Ω \Omega with smooth boundary ∂ Ω \partial \Omega , f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artificial boundary Γ R {\Gamma _R} near infinity, e.g. a sphere of sufficiently large radius R . The intersection of this sphere with Ω C {\Omega ^C} is denoted by Ω R C \Omega _R^C and the given problem is replaced by \[ − Δ u R = f in Ω R C , u R = g on ∂ Ω , ∂ u R ∂ r + 1 r u R = 0 on Γ R . - \Delta {u_R} = f\quad {\text {in}}\;\Omega _R^C,\quad {u_R} = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial {u_R}}}{{\partial r}} + \frac {1}{r}{u_R} = 0\quad {\text {on}}\;{\Gamma _R}. \] This problem is then solved approximately by the finite element method, resulting in an approximate solution u R h u_R^h for each h &gt; 0 h &gt; 0 . In order to obtain a reasonably small error for u − u R h = ( u − u R ) + ( u R − u R h ) u - u_R^h = (u - {u_R}) + ({u_R} - u_R^h) , it is necessary to make R large. This necessitates the solution of a large number of linear equations, so that this method is often not very good when a uniform mesh size h is employed. It is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by C h − 3 C{h^{ - 3}} with C independent of h and R .

Supraconvergent cell-centered scheme for two dimensional elliptic problems

As an optical molecular imaging modality, fluorescence molecular tomography (FMT) can monitor the activities of organisms in vivo at the molecular and cellular levels. However, the recovered image quality is affected by mesh voxel when the finite element method is utilized to recover the fluorescence probe. The target localization is likely to deviate from the actual target under the coarse mesh, but using the fine mesh will increase the number of unknowns, which makes the computational burden heavier and further aggravate the ill-posedness. To solve the problem, a reconstruction strategy using a non-uniform mesh for FMT is developed in this paper. The numerical experiment and physical experiment validated that the strategy is capable and effective for FMT.