Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem

Local defect correction techniques : analysis and application to combustion

We analyze a special case of the Local Defect Correction (LDC) method introduced in [4]. We restrict ourselves to finite difference discretizations of elliptic boundary value problems. The LDC method uses the discretization on a uniform global coarse grid and on one or more uniform local fine grids for approximating the continuous solution. We prove that this LDC method can be seen as an iterative method for solving an underlying composite grid discretization. This result makes it possible to explain important properties of the LDC method, e.g. concerning the size of the discretization error. Furthermore, the formulation of LDC as an iterative solver for a given composite grid problem makes it possible to prove a close correspondence between LDC and the Fast Adaptive Composite grid (FAC) method from [8–10].

A finite volume local defect correction method for solving the transport equation

For elliptic problems a local defect correction method is described. A basic (global) discretization is improved by a local discretization defined in a subdomain. The convergence rate of the local defect correction iteration is proved to be proportional to a certain positive power of the step size. The accuracy of the converged solution can be described. Numerical examples confirm the theoretical results. We discuss multi-grid iterations converging to the same solution.The local defect correction determines a solution depending on one global and one or more local discretizations. An extension of this approach is the domain decomposition method, where only (overlapping) local problems are combined. Such a combination of local subproblems can be solved efficiently by a multi-grid iteration. We describe a multi-grid variant that is suited for the use of parallel processors.KeywordsLocal ProblemDomain DecompositionMultigrid MethodDomain Decomposition MethodInterior BoundaryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Dionne & Golubitsky [10] consider the classification of planforms bifurcating (simultaneously) in scalar partial differential equations that are equivariant with respect to the Euclidean group in the plane. In particular, those planforms corresponding to isotropy subgroups with one-dimensional fixed-point space are classified. Many important Euclidean-equivariant systems of partial differential equations essentially reduce to a scalar partial differential equation, but this is not always true for general systems. We extend the classification of [10] obtaining precisely three planforms that can arise for general systems and do not exist for scalar partial differential equations. In particular, there is a class of one-dimensional ‘pseudoscalar’ partial differential equations for which the new planforms bifurcate in place of three of the standard planforms from scalar partial differential equations. For example, the usual rolls solutions are replaced by a nonstandard planform called anti-rolls. Scalar and pseudoscalar partial differential equations are distinguished by the representation of the Euclidean group.

Adaptive gridding methods are of fundamental importance both for industry and academia. As one of the computing methods, the Boundary Element Method (BEM) is used to simulate problems whose fundamental solutions are available. The method is usually characterised as constant elements BEM or linear elements BEM depending on the type of interpolation used at the elements. Its popularity is steadily growing because of its advantages over other numerical methods most important of which being that it only involves obtaining data at the boundary and computation of the solution in the domain is merely a case of post processing through the use of an identity. This results in the reduction of the problem dimension by unity. Although there is a reduction in dimension when we use BEM, the method usually results in full matrices which can be expensive to solve. This makes the method costly for problems that require very fine grids like those with hot spots as in the example of impressed current cathodic protection systems (ICCP). The BEM is a global method in nature in that the solution in one node depends on the solutions in all the other nodes of the grid. Hence an error in one node can pollute the solution in all the other nodes. In this thesis, we first focus our attention on defining and studying both the local and global errors for the BEM. This is not a completely new study as the literature suggests, however, our approach is different. We use the basic foundations of the the method to define the errors. Since the method is a global method, first we use the interpolation error on each element to define what we have called a sublocal error. Then using the sublocal error we have defined the local error. Understanding the local errors enabled us study the global error. Theoretical and numerical results show that these errors are second order in grid size for both the constant and linear element cases. Then, having explored errors, we study a method for adaptive grid refinement for the BEM. Rather than using a truly nonuniform grid, we present a method called local defect correction (LDC) that is based on local uniform grid refinement. This method is already developed and documented for other numerical methods such as finite difference and finite volume methods but not for BEM. In the LDC method, the discretization on a composite grid is based on a combination of standard discretizations on several uniform grids with different grid sizes that cover different parts of the domain. At least one grid, the uniform global coarse grid, should cover the entire boundary. The size of the global coarse grid is chosen in agreement with the relatively smooth behaviour of the solution outside the hot spots. Then several uniform local fine grids each of which covers only a (small) part of the boundary are used in the hot spots. The grid sizes of the local grids are chosen in agreement with the behaviour of the continuous solution in that part of the boundary. The LDC method is an iterative process whereby a basic global discretisation is improved by local discretisations defined in subdomains. The update of the coarse grid solution is achieved by adding a defect correction term to the right hand side of the coarse grid problem. At each iteration step, the process yields a discrete approximation of the continuous solution on the corresponding composite grid. We have shown how this discretisation can be achieved for the BEM. We apply the discretisation to an academic example to demonstrate its implementation and later show how to use it for an application as the ICCP system. The results show that it is a cheaper method than solving on a truly composite grid and converges in a single step.

A new type of iteration method is proposed in this paper to solve the Steklov eigenvalue problem by the finite element method. In this scheme, solving the Steklov eigenvalue problem is transformed into a series of solutions of boundary value problems on multilevel meshes by the multigrid method and solutions of the Steklov eigenvalue problem on the coarsest mesh. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The computational work of this new scheme for the Steklov eigenvalue problem can reach the same optimal order as the solution of the corresponding boundary value problem. Therefore, an improvement of efficiency for the Steklov eigenvalue solving method can be achieved. Some numerical experiments are presented to validate the efficiency of the new method.

In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.

This paper is concerned with the convergence analysis of the local defect correction (LDC) method for diffusion equations. We derive a general expression for the iteration matrix of the method. We consider the model problem of Poisson's equation on the unit square and use standard five-point finite difference discretizations on uniform grids. It is shown via both an upper bound for the norm of the iteration matrix and numerical experiments, that the rate of convergence of the LDC method is proportional to H 2 with H the grid size of the global coarse grid.

One of the major problems in enhancing the specific work output and efficiency in gas turbines is the maximum possible value of the turbine inlet temperature due to blade material properties. To increase this maximum, turbine blades need to be cooled (internal or external), which is usually done by compressor air. Based on its high cooling efficiency, film cooling is one of the major cooling techniques used, especially for the hottest blades. In film cooling cold air is injected into the boundary layer through small nozzles in the blade surface. Impingement of the jets into the (laminar) boundary layer flow is essentially three-dimensional. The collision of the laminar jet with the boundary layer flow produces a local turbulent shear layer and changes the local heat transfer to the blade (when poorly constructed it may even increase the local heat transfer). In this project we have studied local grid refinement methods and their application to flow problems in general and to air film cooling in particular. Local defect correction (LDC) is an iterative method for solving pure boundary value or initial-boundary value problems on composite grids. It is based on using simple data structures and simple discretization stencils on uniform or tensor-product grids. Fast solution techniques exist for solving the system of equations resulting from discretization on a structured grid. We have combined the standard LDC method with high order finite differences by using a new strategy of defect calculation. Numerical results prove high accuracy and fast convergence of the proposed method. We made a review of boundary conditions for compressible flows. Since we would like to use local grid refinement for such flow problems, we studied the spreading of an acoustic pulse. For this model problem we introduced local grid refinement and made a series of tests in order to see if the artificial boundary conditions introduced for the local fine grid cause any reflections of the acoustic waves. The numerical techniques developed have been used to study film cooling. Because this problem concerns the interaction between a main flow and a jet, we also propose a domain decomposition algorithm in order to supply proper boundary conditions for the cooling jet. This domain decomposition combines a structured DNS flow solver for the problem of interest with an unstructured solver for the flow in the cooling nozzle. Additionally we implemented local grid refinement for the flow problem to save computational costs.

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

In this paper, we consider a L∞ functional derivative estimate for the first spatial derivatives of bounded classical solutions u:RN×[0,T]→R to the Cauchy problem for scalar second order semi-linear parabolic partial differential equations with a continuous nonlinearity f:R→R and initial data u0:RN→R, of the form,maxi=1,…,N⁡(supx∈RN⁡|uxi(x,t)|)≤Ft(f,u0,u)∀t∈[0,T]. Here Ft:At→R is a functional as defined in §1 and x=(x1,x2,…,xn)∈RN. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence (fn,0,u(n)), where for each n∈N, u(n):RN×[0,T]→R is a solution to the Cauchy problem with zero initial data and nonlinearity fn:R→R, and for which there exists α>0 such thatmaxi=1,…,N⁡(supx∈RN⁡|uxi(n)(x,T)|)≥α, whilstlimn→∞⁡(inft∈[0,T]⁡(maxi=1,…,N⁡(supx∈RN⁡|uxi(n)(x,t)|)−Ft(fn,0,u(n))))=0.

This paper discusses the nonconforming rotated Q1 finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective.

ABSTRACTThis paper introduces a kind of parallel multigrid method for solving Steklov eigenvalue problem based on the multilevel correction method. Instead of the common costly way of directly solving the Steklov eigenvalue problem on some fine space, the new method contains some boundary value problems on a series of multilevel finite element spaces and some steps of solving Steklov eigenvalue problems on a very low dimensional space. The linear boundary value problems are solved by some multigrid iteration steps. We will prove that the computational work of this new scheme is truly optimal, the same as solving the corresponding linear boundary value problem. Besides, this multigrid scheme has a good scalability by using parallel computing technique. Some numerical experiments are presented to validate our theoretical analysis.

A DRBEM approximation of the Steklov eigenvalue problem