Wavelet methods for PDEs — some recent developments

On Application of Fast and Adaptive Periodic Battle–Lemarie Wavelets to Modeling of Multiple Lossy Transmission Lines

A streamlined upwind Petrov–Galerkin (SUPG)–stabilized finite-element discretization has been implemented as a library into the FUN3D unstructured-grid flow solver. Motivation for the selection of this methodology is given, details of the implementation are provided, and the discretization for the interior scheme is verified for linear and quadratic elements by using the method of manufactured solutions. A methodology is also described for capturing shocks, and simulation results are compared with the finite-volume formulation that is currently the primary method employed for routine engineering applications. The finite-element methodology is demonstrated to be more accurate than the finite-volume technology, particularly on tetrahedral meshes where the solutions obtained using the finite-volume scheme can suffer from adverse effects caused by bias in the grid. Although no effort has been made to date to optimize computational efficiency, the finite-element scheme is competitive with the finite-volume scheme in terms of computer time to reach convergence.

A Streamlined Upwind Petrov-Galerkin (SUPG) stabilized finite-element discretization has been implemented as a library into the FUN3D unstructured-grid flow solver. Motivation for the selection of this methodology is given, details of the implementation are provided, and the discretization for the interior scheme is verified for linear and quadratic elements by using the method of manufactured solutions. A methodology is also described for capturing shocks, and simulation results are compared to the finite-volume formulation that is currently the primary method employed for routine engineering applications. The finite-element methodology is demonstrated to be more accurate than the finite-volume technology, particularly on tetrahedral meshes where the solutions obtained using the finite-volume scheme can suffer from adverse effects caused by bias in the grid. Although no effort has been made to date to optimize computational efficiency, the finite-element scheme is competitive with the finite-volume scheme in terms of computer time to reach convergence.

In recent years, there has been gratifying progress in the development of domain decomposition algorithms for symmetric and nonsymmetric elliptic problems and even some indefinite problems. Many methods possess the attractive property that the convergence rate is optimal, i.e., independent of the size of the discrete problem and of the number of subdomains, or within a polylog factor of optimal. There is, in comparison, relatively little in the domain decomposition literature on hyperbolic problems. Quarteroni [8, 9] used nonoverlapping domain decomposition methods based on the spectral collocation approximation on systems of conservation laws. Gastaldi and Gastaldi [5, 6] set up a nonoverlapping domain decomposition scheme based on the finite element approximation for the transport equation. These contributions establish the boundary operators that lead to well-posed decoupled problems, which can then be discretized and solved by standard means. Our interests in this paper are rather different. We examine overlapping domain decomposition preconditioners, and leave the original global discretization fully in tact. Rather than deriving interface conditions that lead to decomposed solutions that are mathematically equivalent (to within some specified discretization tolerance) to the solutions of the undecomposed problem, we derive an approximate inverse that can be applied in a concurrent manner, subdomain-by-subdomain, and that effectively preconditions the original undecomposed operator, whose action is already trivial to apply in the same concurrent manner. There seem to have been to date no such additive or multiplicative Schwarz preconditioners leading to optimal convergence rates for hyperbolic equations. Based on the standard Galerkin method [4] an ASM algorithm is formulated. The preconditioned problems are solved by the GMRES method. The convergence

DIn this chapter, we present some of the major results that have been achieved in the context of the DFG-SPP project Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Problems. This project has been concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains as well as with related inverse parameter identification problems. One crucial step has been the design of an efficient forward solver. We employed a spatially adaptive wavelet Rothe scheme. The resulting elliptic subproblems have been solved by adaptive wavelet Galerkin schemes based on generalized tensor wavelets that realize dimension-independent approximation rates. In this chapter, we present the construction of these new tensor bases and discuss some numerical experiments.

We consider two types of inverse boundary problems related to the Laplace equation in planar double connected domains. The first one consists in the determining the Cauchy data on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the function (or the normal derivative) is known and, additionally, on a portion of this exterior boundary the normal derivative (or the function) is also given. For this linear inverse problem we propose a direct boundary integral approach in combination with Tikhonov regularization for stable determination of the Cauchy data on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green's functions. We outline an effective way of discretizing these boundary integral equations based on the Nystrom method and trigonomtric approximations. The second set of inverse problems is the reconstruction of the interior curve from the given Cauchy data of a harmonic function on the exterior boundary. With the help of Green's function and potential theory this non-linear boundary problem is reduced to the system of non-linear boundary integral equations. We develop three iterative algorithms for its numerical solution. Full discretization of the linearized systems is realized by a trigonometric quadrature method. Due to the inherited ill-possedness in the obtained system of linear equations we apply the Tikhonov regularization. Numerical results show that the proposed approach gives good accuracy of reconstructions with economical computational cost.

We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.

A priori error estimates are derived for several Galerkin procedures. For a nonlinear parabolic initial-boundary value problem, optimal rate L2-error estimates are derived for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods. For a nonlinear second order hyperbolic initial-boundary value problem, optimal rate L2-error estimates are derived for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method. 1. Introduction. In this paper we derive a priori error estimates for several Galerkin procedures. In ?3 we consider a nonlinear parabolic initial-boundary value problem, and we derive error estimates for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods of (5); these methods were formulated in (5), but in the central difference methods proofs were given only for the case in which the elliptic part was linear. In ?4 we consider a nonlinear second order hyperbolic initial-boundary value problem, and we derive error estimates for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method of (5); these methods are the obvious extensions of the methods of (5) to the nonlinear case. In ?2 we present certain notation, assumptions, and known results which are common to the development of ? 3 and 4. Section 5 contains some computational results.

Rotational test spaces for a fully-implicit FVM and FEM for the DNS of fluid-particle interaction

An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

In this chapter we introduce some examples of nonlinear hyperbolic problems. We will point out some characteristic properties of such problems, most notably their ability to generate discontinuous solutions also in the case of continuous initial and boundary data. The numerical approximation of these problems is far from easy. Here we will simply limit ourselves to point out how finite difference and finite element schemes can be applied in the case of one-dimensional equations.

Anticandical antibodies in the vaginal discharge and sera of patients with vaginal candidiasis

Nonlinear analysis of Reissner's plate by the variational approaches and boundary element methods

Eddy current nondestructive evaluation (NDE) of airframe structures involves the detection of electromagnetic field irregularities due to non-conducting inhomogeneities in an electrically conducting material. Usually, the eddy current NDE problem can be formulated by the boundary integral equations (BIE) and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The fast multipole method (FMM) is a well-established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large-scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N) operations to compute the system of equations and another O(N) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N). Moreover, several approaches have been proposed for the field calculation in the presence of flaws in three dimensional NDE; however, seldom work has been done in applying efficient methods to seek rapid solution in eddy current NDE simulation. As elaborated in the dissertation, we introduce a fast multipole BIE method for two-dimensional diffusive scalar problem and an efficient BIE method for three-dimensional eddy current NDE. Firstly, we work with the two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometry. We describe the FMM acceleration procedure of the BIE method and its features briefly, explaining that the FMM is not only efficient in meshing complicated geometries, accurate for solving singular fields or

Numerical simulations of flow problems with moving boundaries commonly require the solution of the fluid equations on unstructured and deformable dynamic meshes. In this paper, we present a unified theory for deriving Geometric Conservation La.ws (GCLs) for such problems. We consider several popular discretization methods for the spatial approximation of the flow equations including the Arbitrary Lagrangian-Eulerian (ALE) finite volume and finite element schemes, and space-time stabilized finite element formulations. We show that, except for the case of the space-time discretization method, the GCLs impose import ant constraints on the algorithms employed for time-integrating the semi-discrete equations governing the fluid and dynamic mesh motions. We address the impa.ct of theses constraints on the solution of coupled aeroelastic problems, and highlight the importance of the GCLs with an illustration of their effect on the computation of the transient aeroelastic response of a flat panel in transonic flow.