Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms.     

Solutions of symmetry‐constrained least‐squares problems

In this paper, two new matrix‐form iterative methods are presented to solve the least‐squares problem: and matrix nearness problem: where matrices and are given; ??₁ and ??₂ are the set of constraint matrices, such as symmetric, skew symmetric, bisymmetric and centrosymmetric matrices sets and S_XY is the solution pair set of the minimum residual problem. These new matrix‐form iterative methods have also faster convergence rate and higher accuracy than the matrix‐form iterative methods proposed by Peng and Peng (Numer. Linear Algebra Appl. 2006; 13 : 473–485) for solving the linear matrix equation AXB+CYD=E. Paige's algorithms, which are based on the bidiagonalization procedure of Golub and Kahan, are used as the framework for deriving these new matrix‐form iterative methods. Some numerical examples illustrate the efficiency of the new matrix‐form iterative methods. Copyright © 2008 John Wiley & Sons, Ltd.     

A moving pseudo‐boundary method of fundamental solutions for void detection

We propose a new moving pseudo‐boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least‐squares problem. This approach addresses the major problem of locating the pseudo‐boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems

The integral equations of acoustic and electromagnetic scattering generate large dense systems of linear equations. These systems are efficiently solved with iterative methods where the matrix-vector multiplication is computed using a special fast method, such as the fast Fourier transform or the fast multipole method (FMM). In this paper, the so called diagonal forms of the translation operators for the fast multipole method are derived starting from integral representations of certain special functions. Error analysis of the FMM is given, considering both the truncation error of potential expansions and the errors from the use of numerical integration in the diagonal translation theorem. The implications of the error bounds on the FMM algorithm are discussed.This work has been financially supported by the Jenny and Antti Wihuri Foundation and by the Cultural Foundation of Finland.     

Parallel split least‐squares mixed finite element method for parabolic problem

Based on overlapping domain decomposition, a new class of parallel split least‐squares (PSLS) mixed finite element methods is presented for solving parabolic problem. The algorithm is fully parallel. In the overlapping domains, the partition of unity is applied to distribute the corrections reasonably, which makes that the new method only needs one or two iteration steps to reach given accuracy at each time step while the classical Schwarz alternating methods need many iteration steps. The dependence of the convergence rate on the spacial mesh size, time increment, iteration times, and subdomains overlapping degree is analyzed. Some numerical results are reported to confirm the theoretical analysis.     

Finite element solutions for three‐dimensional elliptic boundary value problems on unbounded domains

The finite element (FE) solutions of a general elliptic equation ?div([a_ij] ??u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R ³, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω_∞, so that the remaining part Ω_B = Ω\Ω _∞ is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γ_a = Ω _∞ ∩ Ω _B. In this article, instead of discarding an unbounded subdomain Ω_∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babu?ka and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On greedy randomized coordinate descent methods for solving large linear least‐squares problems

For solving large scale linear least‐squares problem by iteration methods, we introduce an effective probability criterion for selecting the working columns from the coefficient matrix and construct a greedy randomized coordinate descent method. It is proved that this method converges to the unique solution of the linear least‐squares problem when its coefficient matrix is of full rank, with the number of rows being no less than the number of columns. Numerical results show that the greedy randomized coordinate descent method is more efficient than the randomized coordinate descent method.     

A high‐order compact ADI method for solving three‐dimensional unsteady convection‐diffusion problems

We derive a high‐order compact alternating direction implicit (ADI) method for solving three‐dimentional unsteady convection‐diffusion problems. The method is fourth‐order in space and second‐order in time. It permits multiple uses of the one‐dimensional tridiagonal algorithm with a considerable saving in computing time and results in a very efficient solver. It is shown through a discrete Fourier analysis that the method is unconditionally stable in the diffusion case. Numerical experiments are conducted to test its high order and to compare it with the standard second‐order Douglas‐Gunn ADI method and the spatial fourth‐order compact scheme by Karaa. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Numerical solution of diffraction problems by a least‐squares finite element method

Consider the diffraction of a time‐harmonic wave incident upon a periodic (grating) structure. Under certain assumptions, the diffraction problem may be modelled by a Helmholtz equation with transparent boundary conditions. In this paper, the diffraction problem is formulated as a first‐order system of linear equations and solved by a least‐squares finite element method. The method follows the general minus one norm approach of Bramble, Lazarov, and Pasciak. Our computational experiments indicate that the method is accurate with the optimal convergence property, and it is capable of dealing with complicated grating structures. Copyright © 2000 John Wiley & Sons, Ltd.     

Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems

The traditional method of fundamental solutions (MFS) based on the “global” boundary discretization leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. In this study, a localized version of the MFS (LMFS) is proposed for the large-scale modeling of two-dimensional (2D) elasticity problems. In the LMFS, the whole analyzed domain can be divided into small subdomains with a simple geometry. To each of the subdomain, the traditional MFS formulation is applied and the unknown coefficients on the local geometric boundary can be calculated by the moving least square method. The new method yields a sparse and banded matrix system which makes the method very attractive for large-scale simulations. Numerical examples with up to 200,000 unknowns are solved successfully using the developed LMFS code.     

New a priori analysis of first‐order system least‐squares finite element methods for parabolic problems

We provide new insights into the a priori theory for a time‐stepping scheme based on least‐squares finite element methods for parabolic first‐order systems. The elliptic part of the problem is of general reaction‐convection‐diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a nonsymmetric bilinear form, although the main bilinear form corresponding to the least‐squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the L² norm. Numerical experiments are presented which confirm our theoretical findings.     

Numerical analysis of the method of fundamental solution for harmonic problems in annular domains

In this study, we investigate the application of the method of fundamental solutions (MFS) to the Dirichlet problem for Laplace's equation in an annular domain. We examine the properties of the resulting coefficient matrix and its eigenvalues. The convergence of the method is proved for analytic boundary data. An efficient matrix decomposition algorithm using fast Fourier transforms (FFTs) is developed for the computation of the MFS approximation. We also tested the algorithm numerically on several problems confirming the theoretical predictions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Image solutions for boundary value problems without sources

In this paper, we employ the image method to solve boundary value problems in domains containing circular or spherical shaped boundaries free of sources. two and threeD problems as well as symmetric and anti-symmetric cases are considered. By treating the image method as a special case of method of fundamental solutions, only at most four unknown strengths, distributed at the center, two locations of frozen images and one free constant, need to be determined. Besides, the optimal locations of sources are determined. For the symmetric and anti-symmetric cases, only two coefficients are required to match the two boundary conditions. The convergence rate versus number of image group is numerically performed. The differences of the image solutions between 2D and 3D problems are addressed. It is found that the 2D solution in terms of the bipolar coordinates is mathematically equivalent to that of the simplest MFS with only two sources and one free constant. Finally, several examples are demonstrated to see the validity of the image method for boundary value problems.     

The Trefftz method using fundamental solutions for biharmonic equations

In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations. The bounds of errors are derived for the MFS with Almansi’s fundamental solutions (denoted as the MAFS) in bounded simply connected domains. The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively. The stability analysis of the MAFS is also made for circular domains. Numerical experiments are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples. However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points. Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems. Moreover, the numerical solutions by the MAFS with Almansi’s FS are slightly better in accuracy and stability than those by the traditional MFS. Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity.     

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over‐specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45–52) applied to the Cauchy problem for the two‐dimensional modified Helmholtz equation. The two mixed, well‐posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross‐validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two‐dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

In this paper, the steady‐state Oseen viscous flow equations past a known or unknown obstacle are solved numerically using the method of fundamental solutions (MFS), which is free of meshes, singularities, and numerical integrations. The direct problem is linear and well‐posed, whereas the inverse problem is nonlinear and ill‐posed. For the direct problem, the MFS computations of the fluid flow characteristics (velocity, pressure, drag, and lift coefficients) are in very good agreement with the previously published results obtained using other methods for the Oseen flow past circular and elliptic cylinders, as well as past two circular cylinders. In the inverse obstacle problem the boundary data and the internal measurement of the fluid velocity are minimized using the MATLAB^© optimization toolbox lsqnonlin routine. Regularization was found necessary in the case the measured data are contaminated with noise. Numerical results show accurate and stable reconstructions of various star‐shaped obstacles of circular, bean, or peanut cross‐section.     

Superconvergence of the split least‐squares method for second‐order hyperbolic equations

This article studies superconvergence phenomena of the split least‐squares mixed finite element method for second‐order hyperbolic equations. By selecting the least‐squares functional properly, the procedure can be split into two independent symmetric positive definite subprocedures, one of which is for the primitive unknown and the other is for the flux. Based on interpolation operators and an auxiliary projection, superconvergent H¹ error estimates for the primary variable u and L² error estimates for the introduced flux variable σ are obtained under the standard quasiuniform assumptions on finite element partition. A numerical example is given to show the performance of the introduced scheme. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 222‐238, 2014     

An adaptive least‐squares mixed finite element method for the Signorini problem

We present and analyze a least squares formulation for contact problems in linear elasticity which employs both, displacements and stresses, as independent variables. As a consequence, we obtain stability and high accuracy of our discretization also in the incompressible limit. Moreover, our formulation gives rise to a reliable and efficient a posteriori error estimator. To incorporate the contact constraints, the first‐order system least squares functional is augmented by a contact boundary functional which implements the associated complementarity condition. The bilinear form related to the augmented functional is shown to be coercive and therefore constitutes an upper bound, up to a constant, for the error in displacements and stresses in . This implies the reliability of the functional to be used as an a posteriori error estimator in an adaptive framework. The efficiency of the use of the functional as an a posteriori error estimator is monitored by the local proportion of the boundary functional term with respect to the overall functional. Computational results using standard conforming linear finite elements for the displacement approximation combined with lowest‐order Raviart‐Thomas elements for the stress tensor show the effectiveness of our approach in an adaptive framework for two‐dimensional and three‐dimensional Hertzian contact problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 276–289, 2017     

A variational-type method of fundamental solutions for a Cauchy problem of Laplace’s equation

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace’s equation in a simply-connected bounded domain. Based on a global conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a regularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.