A Nonoverlapping Domain Decomposition Method for Orthogonal Spline Collocation Problems

We consider the initial boundary value problem for a fractional evolution integro-differential equation with a weakly singular kernel arising from viscoelastic rods and membranes in a bounded smooth domain. In our numerical scheme, an arbitrary-order orthogonal spline collocation (OSC) method is used for spatial approximation and combined with the Crank-Nicolson (CN) alternating direction implicit (ADI) method for time approximation, and the integral term is approximated via the product integration (PI) rule. The detailed stability and convergence of the CN ADI OSC method in norm are proved. Finally, some numerical simulations are provided to illustrate our theoretical error estimates. In addition, the comparison between the non-uniform mesh and the uniform mesh shows the effectiveness of the non-uniform mesh when the solution is not sufficiently smooth.

A fast solver for the orthogonal spline collocation solution of the biharmonic Dirichlet problem on rectangles

A direct method of solving Neumann problems for Poisson's equation

ABSTRACTIn this paper, a second-order backward differentiation formula (BDF) orthogonal spline collocation (OSC) method with the truncation error of order in time and in space is presented for a kind of partial differential equation (PDE) with memory. The stability and convergence of the BDF OSC method in a new norm are proved. Besides, we provide three numerical experiments to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the BDF OSC method. Numerical experiments also exhibit the optimal error estimates in the , and-norms.

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.

An Alternating Direction Implicit method is analyzed for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to some separable, second order, ...

The Crank–Nicolson orthogonal spline collocation method for one-dimensional parabolic problems with interfaces

A highly accurate algorithm for retrieving the predicted behavior of problems with piecewise-smooth initial data

We examine the use of orthogonal spline collocation for the semi-discreti\-za\-tion of the cubic Schr\"{o}dinger equation and the two-dimensional parabolic equation of Tappert. In each case, an optimal order\(L^2\) estimate of the error in the semidiscrete approximation is derived. For the cubic Schr\"{o}dinger equation, we present the results of numerical experiments in which the integration in time is performed using a routine from a software library.

Simulation of flow and transport in heterogeneous reservoirs containing many wells presents a variety of numerical difficulties. The wells and heterogeneities represent two different types of singularities. In the examples presented, a two dimensional reservoir contains both injection and production wells. Reservoir properties vary in space and are represented as a stochastic process. Flow is simulated by the modified method of characteristics method and hybrids. Areal displacement patterns and production well recovery curves are monitored. A variety of numerical algorithms are applied to the solution of the equations: strongly implicit procedures, alternating direction methods, multi-griding procedures, pre-conditioned conjugate gradient methods, and domain decomposition. The algorithms are applied separately to three test problems involving (1) mean uniform flow through a heterogeneous media, (2) diverging/converging flow between many wells in a homogeneous media, and (3) flow between many wells in a heterogeneous media. Algorithms which work well in the presence of only one type of singularity, for example test problems (1) or (2), fail in the presence of both singularity types (problem 3). The most robust algorithms for solving problem 3 appear to be particular versions of the preconditioned conjugate gradient method with/without domain decomposition. Adaptive gridding methods may offer another potentially successful approach. Another aspect of the simulations is the recognition that small fluctuations of permeability located near a well have much more influence on flow and displacement than the same fluctuation located at the midpoint between wells. This issue is explored from several points of view: gridding, domain decomposition, and pseudo-properties.

A domain decomposition method for biharmonic equation

The Legendre spectral Galerkin method of self-adjoint second order elliptic equations usually results in a linear system with a dense and ill-conditioned coefficient matrix. In this paper, the linear system is solved by a preconditioned conjugate gradient (PCG) method where the preconditioner M is constructed by approximating the variable coefficients with a (T+1)-term Legendre series in each direction to a desired accuracy. A feature of the proposed PCG method is that the iteration step increases slightly with the size of the resulting matrix when reaching a certain approximation accuracy. The efficiency of the method lies in that the system with the preconditioner M is approximately solved by a one-step method based on the incomplete LU factorization technique with no fill-in, denoted by ILU(0). The ILU(0) factorization of \(M\in {\mathbb {R}}^{(N-1)^d\times (N-1)^d}\) can be computed using \({\mathcal {O}}(T^{2d} N^d)\) operations, and the number of nonzeros in the factorization factors is of \({\mathcal {O}}(T^{d} N^d)\), \(d=1,2,3\). A conclusion of the algorithm is to fast solve the resulting system from the Legendre Galerkin spectral method for Poisson equations with Dirichlet boundary conditions, which has a complexity of \({\mathcal {O}}(N^d)\). To further speed up the PCG method, an algorithm is developed for fast matrix-vector multiplications by the resulting matrix of Legendre-Galerkin spectral discretization, without the need to explicitly form it. The complexity of the fast matrix-vector multiplications is of \({\mathcal {O}}(N^d (\log _2 N)^{2})\). In view that T is independent of N in one dimension and is set to be of order \({\mathcal {O}}(\log _2 N)\) in two and three dimensions, the PCG method has a \({\mathcal {O}}(N^d (\log _2 N)^{2d})\) quasi-optimal complexity for a d dimensional domain with \((N-1)^d\) unknows, \(d=1,2,3\). In addition, a fast direct solver for the three-dimensional Poisson equation is developed, which is of \({\mathcal {O}}(N^{3} (\log _2 N)^{2})\) and improves the existing results on the computational complexity. Numerical examples are given to demonstrate the efficiency of proposed preconditioners and the algorithm for fast matrix-vector multiplications.

In this paper we describe, analyse and implement a parallel iterative method for the solution of the steady-state drift diffusion equations governing the behaviour of a semiconductor device in two space dimensions. The unknowns in our model are the electrostatic potential and the electron and hole quasi-Fermi potentials. Our discretisation consists of a finite element method with mass lumping for the electrostatic potential equation and a hybrid finite element with local current conservation properties for the continuity equations. A version of Gummel's decoupling algorithm which only requires the solution of positive definite symmetric linear systems is used to solve the resulting nonlinear equations. We show that this method has an overall rate of convergence which only degrades logarithmically as the mesh is refined. Indeed the (inner) nonlinear solves of the electrostatic potential equation converge quadratically, with a mesh independent asymptotic constant. We also describe an implementation on a MasPar MP-1 data parallel machine, where the required linear systems are solved by the preconditioned conjugate gradient method. Domain decomposition methods are used to parallelise the required matrix-vector multiplications and to build preconditioners for these very poorly-conditioned systems. Our preconditioned linear solves also have a rate of convergence which degrades logarithmically as the grid is refined relative to subdomain size, and their performance is resilient to the severe layers which arise in the coefficients of the underlying elliptic operators. Parallel experiments are given.

O PTIMAL control problems are often solved numerically via direct methods [1]. In recent years, considerable attention has been focused on a class of direct transcription methods called pseudospectral [2–4] or orthogonal collocation [5,6] methods. In a pseudospectral method, a finite basis of global interpolating polynomials is used to approximate the state at a set of discretization points. The time derivative of the state is approximated by differentiating the interpolating polynomial and constraining the derivative to be equal to the vector field at a finite set of collocation points. Although any set of unique collocation points can be chosen, generally speaking, an orthogonal collocation is chosen, i.e., the collocation points are the roots of an orthogonal polynomial (or linear combinations of such polynomials and their derivatives). Because pseudospectral methods are commonly implemented via orthogonal collocation, the terms pseudospectral and orthogonal collocation are interchangeable (thus researchers in onefieldmay use the term pseudospectral [3], whereas others may use the term orthogonal collocation [5]). One advantage to pseudospectral methods is that for smooth problems, pseudospectral methods typically have faster convergence rates than other methods, exhibiting “spectral accuracy” [4]. Pseudospectral methods were traditionally used to solve fluid dynamics problems [3], whereas orthogonal collocation methods were first established in the chemical engineering community [6]. Seminal work in the mathematics of orthogonal collocation methods for optimal control dates back to 1979 [7] and, in recent years, the following orthogonal collocationmethods have risen to prominence: the Legendre pseudospectral method [8], the Chebyshev pseudospectral method [9], the Radau orthogonal collocation method [10,11], and the Gauss pseudospectral method [12]. Within the class of pseudospectral methods, there are two very different and widely used implementation strategies that can be best described as local and global approaches. In a local approach, the time interval is divided into a large number of subintervals called segments or finite elements [6] and a small number of collocation points are usedwithin each segment. The segments are then linked via continuity conditions on the state, the independent variable, and possibly the control. The rationale for using local collocation is that a local method provides so-called local support [13] (i.e., the discretization points are located so that they support the local behavior of the dynamics) and is both computationally simple and efficient. Although local methods have a long history in solving optimal control problems, much of the recent work has shown great success in the application of global collocation [14,15] (i.e., collocation using a global polynomial across the entire time interval). Researchers are also looking into higher-order local methods [16,17], which lie in between global and local methods. In light of the recent results that promote global orthogonal collocation and the long history of the use local collocation, it is important to gain a better understanding as to how these two different philosophies compare in practice. This Note provides a comparison between global and local orthogonal collocation solutions for two optimal control problems.A recently developed orthogonal collocation method called the Gauss pseudospectral method (GPM) [12] is employed in both a global format and a local format. In the global approach, the GPM is implemented on a single segment and the number of collocation points is varied. As a local approach, the GPM is implemented such that a number of equal width segments are varied, while a small fixed number of collocation points are used in each segment. We note that the local application of the GPM is similar to the approaches of [6,10,18]. The results obtained in this research suggest that, except in special circumstances, global orthogonal collocation is preferable to local orthogonal collocation. For the smooth example in this study, the global GPM is much more accurate than the local GPM for a given number of total collocation points. Furthermore, for a desired accuracy, the global approach is computationally more efficient than the local approach in smooth problems. For nonsmooth problems, as in the second example, the local and global approach are quite similar in terms of accuracy.

A scalable and efficient monolithic approach based on the Balancing Domain Decomposition (BDD) method for acoustic fluid-structure interaction problems is developed. The BDD method is a well-known domain decomposition method for non-overlapping sub-domains, which consists of Neumann-Neumann (NN) preconditioning and coarse grid correction. In this study, we derive four types of BDD method, considering two options for NN preconditioning (NN-I and NN-C) and two options for coarse grid correction (CGC-FULL and CGC-DIAG). From the results of numerical experiments, the combination of NN-I and CGC-FULL turns out to be the most efficient scheme, showing fast convergence property irrespective of the number of sub-domains, DOFs of fluid and solid domains, and the added-mass effect of fluid. The combination of NN-I and CGC-DIAG is also expected to be an efficient scheme in some situations in a parallel environment.