A finite element method for second order nonvariational elliptic problems

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of “finite element Hessian” and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasi-linear PDE, all in nonvariational form.

On Global Attractors for a Class of Parabolic Problems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Developments and extensions of Roth's method of double Fourier series with applications to the solution of electromagnetic field problems

The first part of the thesis compares Roth's method with other methods, in particular the method of separation of variables and the finite cosine transform method, for solving certain elliptic partial differential equations arising in practice. In particular we consider the solution of steady state problems associated with insulated conductors in rectangular slots. Roth's method has two main disadvantages namely the slow rate of con­vergence of the double Fourier series and the restrictive form of the allowable boundary conditions. A combined Roth-separation of variables method is derived to remove the restrictions on the form of the boundary conditions and various Chebyshev approximations are used to try to improve the rate of convergence of the series. All the techniques are then applied to the Neumann problem arising from balanced rectangular windings in a transformer window. Roth's method is then extended to deal with problems other than those resulting from static fields. First we consider a rectangular insulated conductor in a rectangular slot when the current is varying sinusoidally with time. An approximate method is also developed and compared with the exact method.The approximation is then used to consider the problem of an insulated conductor in a slot facing an air gap. We also consider the exact method applied to the determination of the eddy-current loss produced in an isolated rectangular conductor by a transverse magnetic field varying sinusoidally with time. The results obtained using Roth's method are critically compared with those obtained by other authors using different methods. The final part of the thesis investigates further the application of Chebyshdev methods to the solution of elliptic partial differential equations; an area where Chebyshev approximations have rarely been used. A poisson equation with a polynomial term is treated first followed by a slot problem in cylindrical geometry.

An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

In this study, we investigate the problem of reconstruction of a stationary temperature field from given temperature and heat flux on a part of the boundary of a semi-infinite region containing an inclusion. This situation can be modelled as a Cauchy problem for the Laplace operator and it is an ill-posed problem in the sense of Hadamard. We propose and investigate a Landweber-Fridman type iterative method, which preserve the (stationary) heat operator, for the stable reconstruction of the temperature field on the boundary of the inclusion. In each iteration step, mixed boundary value problems for the Laplace operator are solved in the semi-infinite region. Well-posedness of these problems is investigated and convergence of the procedures is discussed. For the numerical implementation of these mixed problems an efficient boundary integral method is proposed which is based on the indirect variant of the boundary integral approach. Using this approach the mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing that stable and accurate reconstructions of the temperature field on the boundary of the inclusion can be obtained also in the case of noisy data. These results are compared with those obtained with the alternating iterative method.

An iterative method for a Cauchy problem for the heat equation

An iterative method for reconstruction of the solution to a parabolic initial boundary value problem of second order from Cauchy data is presented. The data are given on a part of the boundary. At each iteration step, a series of well-posed mixed boundary value problems are solved for the parabolic operator and its adjoint. The convergence proof of this method in a weighted L2-space is included.

Numerical solution of parabolic Cauchy problems in planar corner domains

An iterative method for the parabolic Cauchy problem in planar domains having a finite number of corners is implemented based on boundary integral equations. At each iteration, mixed well-posed problems are solved for the same parabolic operator. The presence of corner points renders singularities of the solutions to these mixed problems, and this is handled with the use of weight functions together with, in the numerical implementation, mesh grading near the corners. The mixed problems are reformulated in terms of boundary integrals obtained via discretization of the time-derivative to obtain an elliptic system of partial differential equations. To numerically solve these integral equations a Nyström method with super-algebraic convergence order is employed. Numerical results are presented showing the feasibility of the proposed approach. © 2014 IMACS.

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.

A coupled boundary element/differential equation method formulation for plate-beam interaction analysis

In this work, a new boundary element formulation for the analysis of plate-beam interaction is presented. This formulation uses a three nodal value boundary elements and each beam element is replaced by its actions on the plate, i.e., a distributed load and end of element forces. From the solution of the differential equation of a beam with linearly distributed load the plate-beam interaction tractions can be written as a function of the nodal values of the beam. With this transformation a final system of equation in the nodal values of displacements of plate boundary and beam nodes is obtained and from it, all unknowns of the plate-beam system are obtained. Many examples are analyzed and the results show an excellent agreement with those from the analytical solution and other numerical methods. (C) 2009 Elsevier Ltd. All rights reserved.

A new wavelet-based method for the solution of the population balance equation

A new wavelet-based method for solving population balance equations with simultaneous nucleation, growth and agglomeration is proposed, which uses wavelets to express the functions. The technique is very general, powerful and overcomes the crucial problems of numerical diffusion and stability that often characterize previous techniques in this area. It is also applicable to an arbitrary grid to control resolution and computational efficiency. The proposed technique has been tested for pure agglomeration, simultaneous nucleation and growth, and simultaneous growth and agglomeration. In all cases, the predicted and analytical particle size distributions are in excellent agreement. The presence of moving sharp fronts can be addressed without the prior investigation of the characteristics of the processes. (C) 2001 Published by Elsevier Science Ltd.

Accurate and efficient simulation of multiphase flow in a heterogeneous reservoir by using error estimate and control in the multiscale finite-volume framework

The multiscale finite-volume (MSFV) method is designed to reduce the computational cost of elliptic and parabolic problems with highly heterogeneous anisotropic coefficients. The reduction is achieved by splitting the original global problem into a set of local problems (with approximate local boundary conditions) coupled by a coarse global problem. It has been shown recently that the numerical errors in MSFV results can be reduced systematically with an iterative procedure that provides a conservative velocity field after any iteration step. The iterative MSFV (i-MSFV) method can be obtained with an improved (smoothed) multiscale solution to enhance the localization conditions, with a Krylov subspace method [e.g., the generalized-minimal-residual (GMRES) algorithm] preconditioned by the MSFV system, or with a combination of both. In a multiphase-flow system, a balance between accuracy and computational efficiency should be achieved by finding a minimum number of i-MSFV iterations (on pressure), which is necessary to achieve the desired accuracy in the saturation solution. In this work, we extend the i-MSFV method to sequential implicit simulation of time-dependent problems. To control the error of the coupled saturation/pressure system, we analyze the transport error caused by an approximate velocity field. We then propose an error-control strategy on the basis of the residual of the pressure equation. At the beginning of simulation, the pressure solution is iterated until a specified accuracy is achieved. To minimize the number of iterations in a multiphase-flow problem, the solution at the previous timestep is used to improve the localization assumption at the current timestep. Additional iterations are used only when the residual becomes larger than a specified threshold value. Numerical results show that only a few iterations on average are necessary to improve the MSFV results significantly, even for very challenging problems. Therefore, the proposed adaptive strategy yields efficient and accurate simulation of multiphase flow in heterogeneous porous media.

Boundary identification in the domain of a parabolic partial differential equation

Bakgrunden och inspirationen till föreliggande studie är tidigare forskning i tillämpningar på randidentifiering i metallindustrin. Effektiv randidentifiering möjliggör mindre säkerhetsmarginaler och längre serviceintervall för apparaturen i industriella högtemperaturprocesser, utan ökad risk för materielhaverier. I idealfallet vore en metod för randidentifiering baserad på uppföljning av någon indirekt variabel som kan mätas rutinmässigt eller till en ringa kostnad. En dylik variabel för smältugnar är temperaturen i olika positioner i väggen. Denna kan utnyttjas som insignal till en randidentifieringsmetod för att övervaka ugnens väggtjocklek. Vi ger en bakgrund och motivering till valet av den geometriskt endimensionella dynamiska modellen för randidentifiering, som diskuteras i arbetets senare del, framom en flerdimensionell geometrisk beskrivning. I de aktuella industriella tillämpningarna är dynamiken samt fördelarna med en enkel modellstruktur viktigare än exakt geometrisk beskrivning. Lösningsmetoder för den s.k. sidledes värmeledningsekvationen har många saker gemensamt med randidentifiering. Därför studerar vi egenskaper hos lösningarna till denna ekvation, inverkan av mätfel och något som brukar kallas förorening av mätbrus, regularisering och allmännare följder av icke-välställdheten hos sidledes värmeledningsekvationen. Vi studerar en uppsättning av tre olika metoder för randidentifiering, av vilka de två första är utvecklade från en strikt matematisk och den tredje från en mera tillämpad utgångspunkt. Metoderna har olika egenskaper med specifika fördelar och nackdelar. De rent matematiskt baserade metoderna karakteriseras av god noggrannhet och låg numerisk kostnad, dock till priset av låg flexibilitet i formuleringen av den modellbeskrivande partiella differentialekvationen. Den tredje, mera tillämpade, metoden kännetecknas av en sämre noggrannhet förorsakad av en högre grad av icke-välställdhet hos den mera flexibla modellen. För denna gjordes även en ansats till feluppskattning, som senare kunde observeras överensstämma med praktiska beräkningar med metoden. Studien kan anses vara en god startpunkt och matematisk bas för utveckling av industriella tillämpningar av randidentifiering, speciellt mot hantering av olinjära och diskontinuerliga materialegenskaper och plötsliga förändringar orsakade av “nedfallande” väggmaterial. Med de behandlade metoderna förefaller det möjligt att uppnå en robust, snabb och tillräckligt noggrann metod av begränsad komplexitet för randidentifiering.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

A Nyström method for a boundary value problem arising in unsteady water wave problems

This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace’s equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl., 20, 121–152). We present a Nystr¨om method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.