A comparison of Galerkin,collocation and the method of lines for partial differential equations

A number of numerical methods for parabolic paritial differential equations are discussed, they all reduce a single partial differential equation to a system of ordinary differential equations by discretization in the space variable(s). The discretizations can be finite element approximations (Galerkin and collocation) or finite difference approximations (methods of lines). The performances of the different methods are compared when used in conjunction with a variable time-step ordinary differential equation integrator. Tables of results are given and the suitability of the discretizations is discussecd.     

Boundary element based formulations for crack shape sensitivity analysis

The present paper addresses several BIE-based or BIE-oriented formulations for sensitivity analysis of integral functionals with respect to the geometrical shape of a crack. Functionals defined in terms of integrals over the external boundary of a cracked body and involving the solution of a frequency-domain boundary-value elastodynamic problem are considered, but the ideas presented in this paper are applicable, with the appropriate modifications, to other kinds of linear field equations as well. Both direct differentiation and adjoint problem techniques are addressed, with recourse to either collocation or symmetric Galerkin BIE formulations. After a review of some basic concepts about shape sensitivity and material differentiation, the derivative integral equations for the elastodynamic crack problem are discussed in connection with both collocation and symmetric Galerkin BIE formulations. Building upon these results, the direct differentiation and the adjoint solution approaches are then developed. In particular, the adjoint solution approach is presented in three different forms compatible with boundary element method (BEM) analysis of crack problems, based on the discretized collocation BEM equations, the symmetric Galerkin BEM equations and the direct and adjoint stress intensity factors, respectively. The paper closes with a few comments.     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

Performance evaluation of algorithms for mildly nonlinear elliptic problems

A number of numerical methods for mildly nonlinear elliptic boundary value problems on general domains is presented. The discretization procedures considered are: a fourth-order FFT-type method, collocation using Hermite bicubic splines and Galerkin with linear triangular as well as quadratic quadrilateral isoparametric elements. The linearized collocation and Galerkin equations are solved by various direct methods available in the ELLPACK system. A comparative study of the above equation solvers is presented for different domain geometries and compilers. The evaluation of software for the general mildly nonlinear elliptic equations is performed over 36 instances from a population of 16 parametrized problems with ‘real world’ and ‘mathematical’ behaviour. The performance data suggests that collocation is an effective method for such general problems, while Galerkin with quadratic quadrilateral isoparametric elements is uniformly superior to the one with linear elements.     

Convolution quadrature method‐based symmetric Galerkin boundary element method for 3‐d elastodynamics

The boundary integral equations in 3‐d elastodynamics contain convolution integrals with respect to the time. They can be performed analytically or with the convolution quadrature method. The latter time‐stepping procedure's benefit is the usage of the Laplace‐transformed fundamental solution. Therefore, it is possible to apply this method also to problems where analytical time‐dependent fundamental solutions might not be known. To obtain a symmetric formulation, the second boundary integral equation has to be used which, unfortunately, requires special care in the numerical implementation since it involves hypersingular kernel functions. Therefore, a regularization for closed surfaces of the Laplace‐transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one. Supplementarily, a weakly singular formulation of the Laplace‐transformed elastodynamic double layer potential is presented. This results in a time domain boundary element formulation involving at least only weakly singular integral kernels. Finally, numerical studies validate this approach with respect to different spatial and time discretizations. Further, a comparison with the wider used collocation method is presented. It is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Symmetric collocation BEM/FEM coupling procedure for 2-D dynamic structural–acoustic interaction problems

 This paper presents a symmetric collocation BEM (SCBEM)/FEM coupling procedure applicable to 2-D time domain structural–acoustic interaction problems. The use of symmetry for BEM not only saves memory storage but also enables the employment of efficient symmetric equation solvers, especially for BEM/FEM coupling procedure. Compared with symmetric Galerkin BEM (SGBEM) where double boundary integration should be carried out, SCBEM can reduce significantly the computing cost. Two numerical examples are included to illustrate the effectiveness and accuracy of the proposed method. Received: 2 November 2001 / Accepted: 27 May 2002     

A symmetric Galerkin formulation and dual reciprocity for 2D elastostatics

In this paper, a symmetric Galerkin boundary integral equation including body force terms is presented. The implementation of the dual reciprocity method to transfer the domain integrals to the boundary is presented in the context of the Galerkin formulation. Several numerical examples involving self-weight and centrifugal body forces are studied to demonstrate the efficiency of the method.     

Symmetric Galerkin boundary formulations employing curved elements

Accounts of the symmetric Galerkin approach to boundary element analysis (BEA) have recently been published. This paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency. A new symmetric Galerkin theoretical formulation for both the (harmonic) heat conduction and the (biharmonic) elasticity problem that employs regularized singular and hypersingular boundary integral equations (BIEs) is presented. The novel use of regularized BIEs in the Galerkin context is shown to allow straightforward incorporation of curved, isoparametric elements. A symmetric reusable intrinsic sample point (RISP) numerical integration algorithm is shown to produce a Galerkin (i.e. double) integration strategy that is competitive with its counterpart (i.e. singular) integration procedure in the collocation BEA approach when the time saved in the symmetric equation solution phase is also taken into account. This new formulation is shown to be capable of employing hypersingular BIEs while obviating the requirement of C¹ continuity, a fact that allows the employment of the popular continuous element technology. The behaviour of the symmetric Galerkin BEA method with regard to both direct and iterative equation solution operations is also addressed. A series of example problems are presented to quantify the performance of this symmetric approach, relative to the more conventional unsymmetric BEA, in terms of both accuracy and efficiency. It is concluded that appropriate implementations of the symmetric Galerkin approach to BEA indeed have the potential to be competitive with, if not superior to, collocation-based BEA, for large-scale problems.     

Meshless Galerkin least-squares method

Collocation method and Galerkin method have been dominant in the existing meshless methods. Galerkin-based meshless methods are computational intensive, whereas collocation-based meshless methods suffer from instability. A new efficient meshless method, meshless Galerkin lest-squares method (MGLS), is proposed in this paper to combine the advantages of Galerkin method and collocation method. The problem domain is divided into two subdomains, the interior domain and boundary domain. Galerkin method is applied in the boundary domain, whereas the least-squares method is applied in the interior domain.The proposed scheme elliminates the posibilities of spurious solutions as that in the least-square method if an incorrect boundary conditions are used. To investigate the accuracy and efficiency of the proposed method, a cantilevered beam and an infinite plate with a central circular hole are analyzed in detail and numerical results are compared with those obtained by Galerkin-based meshless method (GBMM), collocation-based meshless method (CBMM) and meshless weighted least squares method (MWLS). Numerical studies show that the accuracy of the proposed MGLS is much higher than that of CBMM and is close to, even better than, that of GBMM, while the computational cost is much less than that of GBMM.Acknowledgements The authors gratefully acknowledge the support of the National Natural Science Foundation of China with grant number 10172052.     

A regularized collocation boundary element method for linear poroelasticity

This article presents a collocation boundary element method for linear poroelasticity, based on the first boundary integral equation with only weakly singular kernels. This is possible due to a regularization of the strongly singular double layer operator, based on integration by parts, which has been applied to poroelastodynamics for the first time. For the time discretization the convolution quadrature method (CQM) is used, which only requires the Laplace transform of the fundamental solution. Furthermore, since linear poroelasticity couples a linear elastic with an acoustic material, the spatial regularization procedure applied here is adopted from linear elasticity and is performed in Laplace domain due to the before mentioned CQM. Finally, the spatial discretization is done via a collocation scheme. At the end, some numerical results are shown to validate the presented method with respect to different temporal and spatial discretizations.     

Frequency domain analysis of interacting acoustic–elastodynamic models taking into account optimized iterative coupling of different numerical methods

In this work, interacting acoustic–elastodynamic models are analyzed by means of an optimized iterative coupling algorithm. In this iterative coupling procedure, each acoustic/elastodynamic sub-domain of the model is solved independently, and the variables at the common interfaces of the sub-domains are successively renewed, until convergence is achieved. A relaxation parameter is introduced in order to ensure and/or speed up the convergence of the iterative analysis, and an expression to compute optimal values for the relaxation parameter is presented. Several numerical methods are considered to discretize the acoustic and elastodynamic sub-domains of the coupled model, and the performance of these different methodologies, in the coupled analysis, is discussed. In this context, the boundary element method and the method of fundamental solutions are applied to model the acoustic sub-domains, whereas the finite element method, the collocation method and the meshless local Petrov–Galerkin method are applied to model the elastodynamic sub-domains. Independent discretizations of the acoustic/elastodynamic sub-domains are allowed, being no matching nodes required along the common interfaces. At the end of the paper, numerical examples are presented, illustrating the performance and potentialities of the adopted procedures.     

An isogeometric collocation method for efficient random field discretization

This paper presents an isogeometric collocation method for a computationally expedient random field discretization by means of the Karhunen-Loève expansion. The method involves a collocation projection onto a finite-dimensional subspace of continuous functions over a bounded domain, basis splines (B-splines) and nonuniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves an exact geometrical representation of many commonly used physical or computational domains and exploits the regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, in the collocation method, the construction of the system matrices for a d-dimensional eigenvalue problem asks for at most d-dimensional domain integrations, as compared with 2d-dimensional integrations required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a huge computational advantage over the existing Galerkin method. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the collocation method for obtaining eigensolutions.     

Analytical integrations in 2D BEM elasticity

In the context of two‐dimensional linear elasticity, this paper presents the closed form of the integrals that arise from both the standard (collocation) boundary element method and the symmetric Galerkin boundary element method. Adopting polynomial shape functions of arbitrary degree on straight elements, finite part of Hadamard, Cauchy principal values and Lebesgue integrals are computed analytically, working in a local coordinate system. For the symmetric Galerkin boundary element method, a study on the singularity of the external integral is conducted and the outer weakly singular integral is analytically performed. Numerical tests are presented as a validation of the obtained results. Copyright © 2001 John Wiley & Sons, Ltd.     

A fully symmetric multi‐zone Galerkin boundary element method

This paper examines the efficient integration of a Symmetric Galerkin Boundary Element Analysis (SGBEA) method with multi‐zone resulting in a fully symmetric Galerkin multi‐zone formulation. In a previous approach, a Galerkin multi‐zone method was developed where the interfacial nodes are assigned degrees of freedom globally so that the displacement and traction continuity across the zonal interfaces are addressed directly. However, the method was only block symmetric. In the present paper, two new approaches are derived. In the first approach, the degrees of freedom for a particular zone are assigned locally, independent of the other zones. The usual linear set of equations, from the symmetric Galerkin approach, are augmented with an additional set of equations generated by the Galerkin form of hypersingular boundary integrals along the interfaces. Zonal continuity is imposed externally through Lagrange's constraints. This approach is also only block symmetric. The second approach derived from the first, uses the continuity constraints at the zonal assembly level to achieve full symmetry. These methods are compared to collocation multi‐zone and an earlier formulation, on two elasticity problems from the literature. It was found that the second method is much faster than the collocation method for medium to large scale problems, primarily due to its complete symmetry. It is also observed that these methods spend marginally more time on integration than the previous Galerkin multi‐zone method but are better suited to parallel processing. Copyright © 1999 John Wiley & Sons, Ltd.     

A symmetric Galerkin multi-zone boundary element formulation

The recent development of the symmetric Galerkin approach to boundary element analysis (BEA) has been demonstrated to be superior to the collocation method for medium to large problems. This fact has been shown in both heat conduction and elasticity. Accounts of collocation multi-zone analysis techniques have also been prevalent in the literature, dealing with multiple boundary integral relations associated with portions of overall objects. This technique results in overall system matrices with a blocked, sparse, but unsymmetric character. It has been shown that multi-zone techniques can produce smaller solution times than a single zone analysis for large problems. These techniques are useful for multi-material problems as well. This paper presents an approach for combining the benefits of both techniques resulting in a Galerkin multi-zone method, that is overall unsymmetric but contains a significant amount of block symmetry. A condensation technique in the multi-zone solver is shown to exploit the symmetry of the Galerkin formulation as well as the blocked sparsity of the multi-zone technique. This method is compared to collocation multi-zone on two elasticity problems from the literature. It is concluded that an appropriate implementation of the symmetric Galerkin multi-zone BEA indeed has the potential to be superior to the collocation based multi-zone BEA, for medium to large-scale elasticity problems. © 1997 John Wiley & Sons, Ltd.     

A boundary integral Trefftz formulation with symmetric collocation

The Galerkin and collocation methods are combined in the implementation of a boundary integral formulation based on the Trefftz method for linear elastostatics. A finite element approach is used in the derivation of the formulation. The domain is subdivided in regions or elements, which need not be bounded, simply connected or convex. The stress field is directly approximated in each element using a complete solution set of the governing Beltrami condition. This stress basis is used to enforce on average, in the Galerkin sense, the compatibility and elasticity conditions. The boundary of each element is, in turn, subdivided into boundary elements whereon the displacements are independently approximated using Dirac functions. This basis is used to enforce by collocation the static admissibility conditions, which reduce to the Neumann conditions as the stress approximation satisfies locally the domain equilibrium condition. The resulting solving system is symmetric and sparse. The coefficients of the structural matrices and vectors are defined either by regular boundary integral expressions or determined by direct collocation of the trial functions.     

A new singular integral equation for the classical crack problem in plane and antiplane elasticity

A new singular integral equation (with a kernel with a logarithmic singularity) is proposed for the crack problem inside an elastic medium under plane or antiplane conditions. In this equation the integral is considered in the sense of a finite-part integral of Hadamard because the unknown function presents singularities of order ?3/2 at the crack tips. The Galerkin and the collocation methods are proposed for the numerical solution of this equation and the determination of the values of the stress intensity factors at the crack tips and numerical results are presented. Finally, the advantages of this equation are also considered.     

A collocation finite element method for potential problems in irregular domains

A potentially powerful numerical method for solving certain boundary value problems is developed. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees-of-freedom per node are used. A subparametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representation of irregular boundaries. It is shown that by taking advantage of the boundary conditions, a minimum number of collocation points can be used. The method is particularly suitable for potential and mass transport problems where a C¹ continuous solution is required. In contrast to the Galerkin approach, it does not require the evaluation of basis function products and numerical integration, also the coefficient matrix contains only about half as many non-zero terms as the corresponding Galerkin coefficient matrix. This results in approximately a 90 per cent reduction in formulation and a 50 per cent reduction in solution operation, as compared with the Galerkin finite element method, for this type of problem. Examples show that the accuracy of the collocation solution is as good as or better than that of the Galerkin solution.     

Aliasing errors due to quadratic nonlinearities on triangular spectral /hp element discretisations

In this paper, consideration is given to how aliasing errors, introduced when evaluating nonlinear products, inexactly affect the solution of Galerkin spectral/hp element polynomial discretisations on triangles. A theoretical discussion is presented of how aliasing errors are introduced by a collocation projection onto a set of quadrature points insufficient for exact integration, and consider interpolation projections to geometrically symmetric ollocation points. The discussion is corroborated by numerica examples that elucidate the key features. The study is first motivated with a review of aliasing errors introduced in one-dimensional spectral-element methods (these results extend naturally to tensor-product quadrilaterals and hexahedra.) Within triangular domains two commonly used expansions are a hierarchical, or modal, expansion based on a rotationally non-symmetric collapsed-coordinate system, and a Lagrange expansion based on a set of rotationally symmetric nodal points. Whilst both expansions span the same polynomial space, the construction of the two bases numerically motivates a different set of collocation points for use in the collocation projection of a nonlinear product. The purpose of this paper is to compare these two collocation projections. The analysis and results show that aliasing errors produced using a collocation projection on the rotationally non-symmetric, collapsed-coordinate system are significantly smaller than those for a collocation projection using the rotationally symmetric nodal points. In the case of the collapsed coordinate projection, if the Gaussian quadrature order employed is less than half the polynomial order of the integrand, then it is possible for the aliasing error to modify the constant mode of the expansion and therefore affect the conservation property of the approximation. However, the use of a collocation projection onto a polynomial expansion associated with a set of rotationally symmetric nodal points within the triangle is always observed to be non-conservative. Nevertheless, the rotationally symmetric collocation will maintain the overall symmetry of the triangular region, which is not typically the case when a collapsed coordinate quadrature projection is used.     

An alternative collocation boundary element method for static and dynamic problems

A collocation boundary element formulation is presented which is based on a mixed approximation formulation similar to the Galerkin boundary element method presented by Steinbach (SIAM J Numer Anal 38:401–413, 2000) for the solution of Laplace’s equation. The method is also applicable to vector problems such as elasticity. Moreover, dynamic problems of acoustics and elastodynamics are included. The resulting system matrices have an ordered structure and small condition numbers in comparison to the standard collocation approach. Moreover, the employment of Robin boundary conditions is easily included in this formulation. Details on the numerical integration of the occurring regular and singular integrals and on the solution of the arising systems of equations are given. Numerical experiments have been carried out for different reference problems. In these experiments, the presented approach is compared to the common nodal collocation method with respect to accuracy, condition numbers, and stability in the dynamic case.