Csaba Gáspár

7005776404

Publications - 29

Method of fundamental solutions formulations for biharmonic problems

Andreas Karageorghis Csaba Gáspár

Publication Name: Engineering Analysis with Boundary Elements

Publication Date: 2025-06-01

Volume: 175

Issue: Unknown

Page Range: Unknown

Description:

We consider various method of fundamental solution (MFS) formulations for the numerical solution of two-dimensional boundary value problems (BVPs) governed by the homogeneous biharmonic equation. The motivation for employing the proposed techniques comes from the corresponding boundary integral representations. These are carefully analyzed in the case the domain of the BVP under consideration is a disk. The results of this analysis detect a potentially troublesome case in one of the proposed MFS approaches. Numerical results confirm the analytical findings for more general domains.

Open Access: Yes

DOI: 10.1016/j.enganabound.2025.106180

A Localized Multi-level Method of Fundamental Solutions for Inhomogeneous Problems

Csaba Gáspár

Publication Name: Lecture Notes in Computational Science and Engineering

Publication Date: 2025-01-01

Volume: 153 LNCSE

Issue: Unknown

Page Range: 345-354

Description:

A localization technique is proposed to solve some elliptic problems based on the classical Schwarz overlapping method. The original problem is converted to solving a sequence of subproblems (which are much less than the original problem) resulting in an iterative method. The local problems are solved by the Method of Fundamental Solutions. The localization technique makes it possible to apply the method to inhomogeneous problems and to more general problems as well. The method can be embedded into a natural multi-level context, which significantly improves the computational efficiency. As a further application, the proposed method is applied to the scattered data interpolation problem.

Open Access: Yes

DOI: 10.1007/978-3-031-86173-4_35

A Localized Method of Fundamental Solutions for the Stokes Equations

Csaba Gáspár

Publication Name: Advances in Science and Technology

Publication Date: 2025-01-01

Volume: 165 AST

Issue: Unknown

Page Range: 31-39

Description:

The Method of Fundamental Solution applied to the Stokes equation is investigated. Instead of using the classical approach, the problem is split into several subproblems defined on much smaller subdomains. Each local problem is solved by the Method of Fundamental Solutions. Having solved the local problems, the approximate solution is updated at the central point of the local subdomain. This results in a Seidel-like iterative method, which mimics the classical overlapping Schwarz method. In contrast to the traditional Method of Fundamental Solutions, the resulting localized method avoids the problem of large, dense and ill-conditioned linear systems of equations, and, at the same time, remains a truly meshfree technique.

Open Access: Yes

DOI: 10.4028/p-SK5Xfk

Application of the localized method of fundamental solutions to heat transfer problems

Csaba Gáspár

Publication Name: Journal of Physics Conference Series

Publication Date: 2024-01-01

Volume: 2766

Issue: 1

Page Range: Unknown

Description:

A localized version of the Method of Fundamental Solutions is applied to the 2D steady heat transfer equation with spatially varying thermal conductivity. Though the corresponding fundamental solution cannot be computed in general, the localization splits the original problem into several subproblems defined on small subdomains; in these subdomains, the subproblems are approximately converted to certain convection-diffusion equations with constant coefficients, so that the Method of Fundamental Solutions is applicable. In each subdomain, the corresponding subproblem is solved separately. This results in an iterative method, which mimics the overlapping (alternating) Schwarz method. Due to its advantageous numerical properties, the technique seems a useful generalization of the Method of Fundamental Solutions. The method is illustrated through numerical examples.

Open Access: Yes

DOI: 10.1088/1742-6596/2766/1/012157

A Localized Version of the Method of Fundamental Solutions in a Multi-level Context

Csaba Gáspár

Publication Name: Periodica Polytechnica Civil Engineering

Publication Date: 2023-01-01

Volume: 67

Issue: 3

Page Range: 716-724

Description:

The Method of Fundamental Solutions is applied to the Laplace equation. Instead of using the traditional approach with external source points and boundary collocation points, the original domain decomposed into a lot of smaller, overlapping subdomains, and the Method of Fundamental Solutions is used to the individual local subdomains. After eliminating the local source points, local schemes are obtained. Instead of constructing a global scheme, the local subproblems are solved sequentially, in an iterative way. This mimics a multiplicative Schwarz method with overlapping subdomains, which assures the convergence of the method. Combining the iteration with a simple Seidel-type method, the resulting iteration is used as a smoothing procedure of a multi-level method. The points belonging to the coarse and fine levels are defined by a quadtree-generated cell system controlled by the boundary of the original domain. The multi-level character of the obtained method makes it possible to reduce the necessary number of iterations, that is, the overall computational cost can be significantly reduced. Moreover, the solution of large and ill-conditioned systems is completely avoided. The method is illustrated through several numerical test examples.

Open Access: Yes

DOI: 10.3311/PPci.21535

Biharmonic Scattered Data Interpolation Based on the Method of Fundamental Solutions

Csaba Gáspár

Publication Name: Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics

Publication Date: 2023-01-01

Volume: 14076 LNCS

Issue: Unknown

Page Range: 485-499

Description:

The two-dimensional scattered data interpolation problem is investigated. In contrast to the traditional Method of Radial Basis Functions, the interpolation problem is converted to a higher order (biharmonic or modified bi-Helmholtz) partial differential equation supplied with usual boundary conditions as well as pointwise interpolation conditions. To solve this fourth-order problem, the Method of Fundamental Solutions is used. The source points, which are needed in the method, are located partly in the exterior of the domain of the corresponding partial differential equation and partly in the interpolation points. This results in a linear system with possibly large and fully populated matrix. To make the computations more efficient, a localization technique is applied, which splits the original problem into a sequence of local problems. The system of local equations is solved in an iterative way, which mimics the classical overlapping Schwarz method. Thus, the problem of large and ill-conditioned matrices is completely avoided. The method is illustrated via a numerical example.

Open Access: Yes

DOI: 10.1007/978-3-031-36027-5_38

Application of quadtrees in the method of fundamental solutions using multi-level tools

Csaba Gáspár

Publication Name: Sema Simai Springer Series

Publication Date: 2020-01-01

Volume: 23

Issue: Unknown

Page Range: 41-57

Description:

The traditional version of the Method of Fundamental Solutions is revisited, which is based on using external sources. The sources are defined in a completely automatic way using a quadtree cell system controlled by the boundary of the domain. This results in a spatial density distribution of sources which decreases rapidly when going far away from the boundary. A similar technique is also proposed, when the sources are located along the boundary (which can be automated easily) and the collocation points are moved into the interior of the domain. In this case, the boundary conditions have to be properly redefined at the inner collocation points, This is done by using boundary-controlled quadtrees again. Both techniques can be embedded in a multi-level context in a natural way. The accuracy of the resulting methods are less than that of the traditional Method of Fundamental Solutions, but it is still acceptable. However, the computational cost is more moderate and the problems of singularity and the extremely ill-conditioned matrices are avoided.

Open Access: Yes

DOI: 10.1007/978-3-030-52804-1_3

A multi-level technique for the Method of Fundamental Solutions without regularization and desingularization

Csaba Gáspár

Publication Name: Engineering Analysis with Boundary Elements

Publication Date: 2019-06-01

Volume: 103

Issue: Unknown

Page Range: 145-159

Description:

The traditional Method of Fundamental Solutions is revisited, based on a special strategy of defining the external source points. Unlike the classical Method of Fundamental Solutions, the sources are categorized into groups; the density of the spatial distribution of the sources decreases rapidly far from the boundary. On each group, the original problem is discretized using the same set of boundary collocation points. Such groups of sources are constructed in a fully automated way by the quadtree/octtree algorithm. The discretized problems are solved in the sense of least squares. A simple multi-level method is built up, using the (conjugate) gradient iteration as a smoothing procedure. The resulting method significantly reduces the computational complexity. Moreover, the problem of evaluation singular integrals as well as the problem of severely ill-conditioned matrices are avoided. The method is generalized to 3D axisymmetric potential problems as well.

Open Access: Yes

DOI: 10.1016/j.enganabound.2019.03.006

A fast and stable multi-level solution technique for the method of fundamental solutions

Csaba Gáspár

Publication Name: Lecture Notes in Computational Science and Engineering

Publication Date: 2019-01-01

Volume: 129

Issue: Unknown

Page Range: 19-42

Description:

The classical form of the Method of Fundamental Solutions is applied. Instead of using a single set of subtly located external sources, a special strategy of defining several sets of external source points is introduced. The sets of sources are defined by the quadtree/octtree subdivision technique controlled by the boundary collocation points in a completely automatic way, resulting in a point set, the density of the spatial distribution of which decreases quickly far from the boundary. The ‘far’ sources are interpreted to form a ‘coarse grid’, while the densely distributed ‘near-boundary’ sources are considered a ‘fine grid’ (despite they need not to have any grid structure). Based on this classification, a multi-level technique is built up, where the smoothing procedure is defined by performing some familiar iterative technique e.g. the (conjugate) gradient method. The approximate solutions are calculated by enforcing the boundary conditions in the sense of least squares. The resulting multi-level method is robust and significantly reduces the computational cost. No weakly or strongly singular integrals have to be evaluated. Moreover, the problem of severely ill-conditioned matrices is completely avoided.

Open Access: Yes

DOI: 10.1007/978-3-030-15119-5_2

The Method of Fundamental Solutions Combined with a Multi-level Technique

Csaba Gáspár

Publication Name: Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics

Publication Date: 2019-01-01

Volume: 11386 LNCS

Issue: Unknown

Page Range: 241-249

Description:

A traditional idea of the Method of Fundamental Solutions is to use some external source points where the fundamental solution should be shifted to. However, the proper definition of the locations of the sources can hardly be performed in an automated way. To circumvent this difficulty, in this paper, the source points defined along the boundary, and the collocation points are shifted to the interior of the domain together with a proper modification of the boundary conditions. Thus, the problem of singularity is avoided. The modified boundary conditions are defined on the basis of the tools of the classical finite difference methods. Several schemes are presented. The schemes can be embedded in a multi-level context in a natural way. The proposed method avoids the computational difficulties due to ill-conditioned matrices and also reduces the computational complexity of the Method of Fundamental Solutions.

Open Access: Yes

DOI: 10.1007/978-3-030-11539-5_26

An implicit Method of Fundamental Solutions using twofold domains

Csaba Gáspár

Publication Name: Engineering Analysis with Boundary Elements

Publication Date: 2017-09-01

Volume: 82

Issue: Unknown

Page Range: 194-201

Description:

A novel meshless method is presented. It is based on completing the original problem with an additional one which is defined in the same domain and connected with the original problem through the boundary only. The boundary conditions are enforced by some internal sources appearing in the supplementary problem only. The fundamental solutions are not used explicitly. The two problems are solved simultaneously using computationally quite efficient multi-level tools. No dense and ill-conditioned linear systems have to be solved. Moreover, no external or boundary sources are to be located. The approach can easily be extended to more general elliptic problems in a natural way, since the corresponding fundamental solution is not needed to be known.

Open Access: Yes

DOI: 10.1016/j.enganabound.2017.06.007

Fast meshless techniques based on the regularized method of fundamental solutions

Csaba Gáspár

Publication Name: Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics

Publication Date: 2017-01-01

Volume: 10187 LNCS

Issue: Unknown

Page Range: 334-341

Description:

A regularized method of fundamental solutions is presented. The method can handle Neumann and mixed boundary conditions as well without using a desingularization technique. Instead, the approach combines the regularized method of fundamental solutions with traditional finite difference techniques based on some inner collocation points located in the vicinity of the boundary collocation points. Nevertheless, the resulting method remains meshless. The method avoids the problem of singularity and can be embedded in a natural multi-level context. The method is illustrated via a numerical example.

Open Access: Yes

DOI: 10.1007/978-3-319-57099-0_36

A regularized multi-level technique for solving potential problems by the method of fundamental solutions

Csaba Gáspár

Publication Name: Engineering Analysis with Boundary Elements

Publication Date: 2015-06-03

Volume: 57

Issue: Unknown

Page Range: 66-71

Description:

The method of fundamental solutions is investigated in the case when the source points are located along the boundary of the domain of the original problem and coincide with the collocation points. The appearing singularities are eliminated by several techniques: by using approximate but continuous fundamental solutions (regularization) and via auxiliary subproblems to avoid the stronger singularities that appear in the normal derivatives of the fundamental solution (desingularization). Both monopole and dipole formulations are investigated. A special iterative solution algorithm is presented, which converts the original (mixed) problem to a sequence of pure Dirichlet and pure Neumann subproblems. The pure subproblems can be handled efficiently by using conjugate gradients. The efficiency is significantly increased by embedding the resulting method in a natural multi-level context. At the same time, the problem of the use of highly ill-conditioned matrices is also avoided.

Open Access: Yes

DOI: 10.1016/j.enganabound.2014.05.002

Regularization and multi-level tools in the method of fundamental solutions

Csaba Gáspár

Publication Name: Lecture Notes in Computational Science and Engineering

Publication Date: 2015-01-01

Volume: 100

Issue: Unknown

Page Range: 145-162

Description:

The Method of Fundamental Solution is applied to potential problems. The source and collocation points are supposed to coincide and are located along the boundary. The singularities due to the singularity of the fundamental solution are avoided by several techniques (regularization and desingularization). Both the monopole and the dipole formulations are investigated. The resulting algebraic systems have advantageous properties provided that pure Dirichlet or pure Neumann boundary condition is prescribed. Otherwise, the original problem is converted to a sequence of pure Dirichlet and pure Neumann subproblems, the solutions of which converge rapidly to the solution of the original mixed problem. The iteration is embedded to a multi-level context in a natural way. Thus, the computational cost can be significantly reduced, and the problem of large and ill-conditioned matrices is also avoided.

Open Access: Yes

DOI: 10.1007/978-3-319-06898-5_8

A regularized method of fundamental solutions for 3D and axisymmetric potential problems

Csaba Gáspár

Publication Name: CMES Computer Modeling in Engineering and Sciences

Publication Date: 2014-01-01

Volume: 101

Issue: 6

Page Range: 365-386

Description:

The Method of Fundamental Solutions (MFS) is investigated for 3D potential problem in the case when the source points are located along the boundary of the domain of the original problem and coincide with the collocation points. This generates singularities at the boundary collocation points, which are eliminated in different ways. The (weak) singularities due to the singularity of the fundamental solution at the origin are eliminated by using approximate but continuous fundamental solution instead of the original one (regularization). The (stronger) singularities due to the singularity of the normal derivatives of the fundamental solution are eliminated by solving special auxiliary subproblems (desingularization). The desingularization idea is similar to a previously published technique and is completely independent of the applied regularization technique. The presented method produces well-conditioned or moderately ill-conditioned matrices in the resulting linear system of algebraic equations, while the accuracy remains acceptable. No boundary mesh structure is needed. The method is generalized to 3D axisymmetric potential problems in a natural way, despite in this case the fundamental solution does not remain a radial function. The use of extremely ill-conditioned matrices is still avoided.

Open Access: Yes

DOI: DOI not available

Some variants of the method of fundamental solutions: Regularization using radial and nearly radial basis functions

Csaba Gáspár

Publication Name: Central European Journal of Mathematics

Publication Date: 2013-08-01

Volume: 11

Issue: 8

Page Range: 1429-1440

Description:

The method of fundamental solutions and some versions applied to mixed boundary value problems are considered. Several strategies are outlined to avoid the problems due to the singularity of the fundamental solutions: the use of higher order fundamental solutions, and the use of nearly fundamental solutions and special fundamental solutions concentrated on lines instead of points. The errors of the approximations as well as the problem of ill-conditioned matrices are illustrated via numerical examples. © 2013 Versita Warsaw and Springer-Verlag Wien.

Open Access: Yes

DOI: 10.2478/s11533-013-0251-7

A regularized Method of Fundamental Solutions without desingularization

Csaba Gáspár

Publication Name: CMES Computer Modeling in Engineering and Sciences

Publication Date: 2013-07-30

Volume: 92

Issue: 1

Page Range: 103-121

Description:

Some regularized versions of the Method of Fundamental Solutions are investigated. The problem of singularity of the applied method is circumvented in various ways using truncated or modified fundamental solutions, or higher order fundamental solutions which are continuous at the origin. For pure Dirichlet problems, these techniques seem to be applicable without special additional tools. In the presence of Neumann boundary condition, however, they need some desingularization techniques to eliminate the appearing strong singularity. Using fundamental solutions concentrated to lines instead of points, the desingularization can be omitted. The method is illustrated via numerical examples. Copyright © 2013 Tech Science Press.

Open Access: Yes

DOI: DOI not available

Regularization techniques for the method of fundamental solutions

Csaba Gáspár

Publication Name: International Journal of Computational Methods

Publication Date: 2013-03-01

Volume: 10

Issue: 2

Page Range: Unknown

Description:

A special regularization method based on higher-order partial differential equations is presented. Instead of using the fundamental solution of the original partial differential operator with source points located outside of the domain, the original second-order partial differential equation is approximated by a higher-order one, the fundamental solution of which is continuous at the origin. This allows the use of the method of fundamental solutions (MFS) for the approximate problem. Due to the continuity of the modified operator, the source points and the boundary collocation points are allowed to coincide, which makes the solution process simpler. This regularization technique is generalized to various problems and combined with the extremely efficient quadtree-based multigrid methods. Approximation theorems and numerical experiences are also presented. © 2013 World Scientific Publishing Company.

Open Access: Yes

DOI: 10.1142/S0219876213410041

Some regularized versions of the method of fundamental solutions

Csaba Gáspár

Publication Name: Lecture Notes in Computational Science and Engineering

Publication Date: 2013-01-21

Volume: 89 LNCSE

Issue: Unknown

Page Range: 181-198

Description:

A powerful method of the solution of homogeneous equations is considered. Using the traditional approach of the Method of Fundamental Solutions, the fundamental solution has to be shifted to external source points. This is inconvenient from computational point of view, moreover, the resulting linear system can easily become severely ill-conditioned. To overcome this difficulty, a special regularization technique is applied. In this approach, the original second-order elliptic problem (a modified Helmholtz problem in the paper) is approximated by a fourth-order multi-elliptic boundary interpolation problem. To perform this boundary interpolation, either the Method of Fundamental Solutions, or a direct multi-elliptic interpolation can be used. In the paper, a priori error estimations are deduced. A numerical example is also presented. © 2013 Springer-Verlag.

Open Access: Yes

DOI: 10.1007/978-3-642-32979-1_12

Meshfree vectorial interpolation based on the generalized stokes problem

Csaba Gáspár

Publication Name: Lecture Notes in Computational Science and Engineering

Publication Date: 2011-01-01

Volume: 79 LNCSE

Issue: Unknown

Page Range: 65-80

Description:

A vectorial interpolation problem is considered. In addition to the interpolation conditions taken at discrete points, a global, divergence-free condition is also prescribed. Utilizing the idea of the multi-elliptic interpolation, the divergencefree interpolation problem is converted to a generalized Stokes problem. To numerically solve this new problem, an Uzawa-type method and the method of fundamental solutions are proposed. In the second method, a linear system with large and dense matrix is to be solved, while in the first method, this problem is avoided. © 2011 Springer-Verlag Berlin Heidelberg.

Open Access: Yes

DOI: 10.1007/978-3-642-16229-9_4

Multi-level meshless methods based on direct multi-elliptic interpolation

Csaba Gáspár

Publication Name: Journal of Computational and Applied Mathematics

Publication Date: 2009-04-15

Volume: 226

Issue: 2

Page Range: 259-267

Description:

A short overview on the direct multi-elliptic interpolation and the related meshless methods for solving partial differential equations is given. A new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain. An error estimation is also derived. To implement the method, quadtree-based multi-level methods are used. The approach avoids the use of large, dense and ill-conditioned matrices and significantly reduces the computational cost. © 2008 Elsevier B.V. All rights reserved.

Open Access: Yes

DOI: 10.1016/j.cam.2008.08.005

Several meshless solution techniques for the stokes flow equations

Csaba Gáspár

Publication Name: Computational Methods in Applied Sciences

Publication Date: 2009-01-01

Volume: 11

Issue: Unknown

Page Range: 141-158

Description:

The permanent 2D Stokes flow is considered. The applied solution technique is the classical pressure correction method, which converts the original problem to a sequence of Poisson equations. These Poisson equations are discretized and solved in a meshless way, using local interpolation based on radial basis functions. Further improvement can be achieved by using a direct multi-elliptic approach instead of local interpolation, which results in re-globalized, quadtree-based schemes. The number of unknowns can be reduced by applying the method of fundamental solutions. A special regularization technique is introduced which uses higher order fundamental solutions without singularities. This regularization is combined with the direct multi-elliptic interpolation, which significantly reduces the computational cost and makes it possible to avoid the use of dense and ill-conditioned matrices.

Open Access: Yes

DOI: 10.1007/978-1-4020-8821-6_9

Boundary interpolation vs boundary elements: Theory and some applications

Csaba Gáspár

Publication Name: International Series on Advances in Boundary Elements

Publication Date: 2004-10-11

Volume: 19

Issue: Unknown

Page Range: 143-152

Description:

Domain and boundary type meshless methods based on the Direct Multi-Elliptic Interpolation Method are presented. The approach is equivalent to a special RBF-method but completely avoids the solution of large, full and ill-conditioned systems, thus, the computational cost is significantly reduced. The method is illustrated through the example of the usual Poisson problem. Both Dirichlet and Neumann boundary conditions are investigated. The domain version of the method results in particular solutions, while the boundary version can be applied to solve homogeneous problems. Along Neumann boundaries, either off-boundary points can be introduced or a boundary reconstruction technique based on boundary interpolation can be applied. Some further possible applications are also outlined.

Open Access: Yes

DOI: DOI not available

A meshless polyharmonic-type boundary interpolation method for solving boundary integral equations

Csaba Gáspár

Publication Name: Engineering Analysis with Boundary Elements

Publication Date: 2004-01-01

Volume: 28

Issue: 10 SPEC. ISS.

Page Range: 1207-1216

Description:

A boundary interpolation technique is introduced based on multi-elliptic partial differential equations. The interpolation problem is converted to a special higher order partial differential equation which is completely independent of the geometry of the original problem. Based on this interpolation method, meshless methods are constructed for the 2D Laplace-Poisson equation. The presented approach makes it possible to avoid solving large and dense interpolation equations. The auxiliary higher order partial differential equation is solved by robust, quadtree-based multi-level methods. The results can be easily generalized to 3D problems as well. © 2004 Elsevier Ltd. All rights reserved.

Open Access: Yes

DOI: 10.1016/j.enganabound.2003.04.001

A meshless boundary element technique based on multi-level iterated Helmholtz-type interpolation

Csaba Gáspár

Publication Name: International Series on Advances in Boundary Elements

Publication Date: 2002-12-01

Volume: 13

Issue: Unknown

Page Range: 105-114

Description:

A special scattered data interpolation technique is introduced. The interpolation problem is converted to a higher order auxiliary PDE, typically to an iterated Laplace or Helmholtz equation. To solve this PDE, robust multilevel methods are used which are based on a quadtree/octtree subdivision algorithm. Thus, the solution of large interpolation equations are avoided. Using this interpolation method, meshless techniques are constructed which require a set of boundary points only, without any structure. Theoretical results as well as numerical examples are also presented.

Open Access: Yes

DOI: DOI not available

Multi-level biharmonic and bi-Helmholtz interpolation with application to the boundary element method

Csaba Gáspár

Publication Name: Engineering Analysis with Boundary Elements

Publication Date: 2000-01-01

Volume: 24

Issue: 7-8

Page Range: 559-573

Description:

The scattered data interpolation problem is investigated. Instead of the direct use of radial basis functions, the interpolation function is sought as a solution of a higher order partial differential equation supplied with the interpolation equations as special boundary conditions. In this paper the methods based on the biharmonic and the bi-Helmholtz equations are analyzed. The interpolation problem is reformulated in variational forms. Existence and uniqueness theorems are proved in Sobolev spaces. The approximation properties of this interpolation are also investigated. A representation theorem is proved which shows the similarity to the method of radial basis functions based on the fundamental solution of the applied partial differential operator. To solve the appearing biharmonic/bi-Helmholtz equation, a multi-level method is presented which is based on a quadtree/octtree cell system generated by the interpolation points. It is shown that the overall computational cost of the presented method is much less than that of the traditional method of radial basis functions. The method makes it possible to avoid the solution of large, fully populated and often ill-conditioned systems of linear equations. Finally, some applications to solving partial differential equations are outlined. The biharmonic/bi-Helmholtz interpolation technique immediately defines a grid-free method, but can be combined with the boundary element method as well. A possible application in the dual reciprocity method is also presented.

Open Access: Yes

DOI: 10.1016/S0955-7997(00)00036-9

On the negative weighting factors in the Muskingum-Cunge scheme

Sàndor Szél Csaba Gáspár

Publication Name: Journal of Hydraulic Research

Publication Date: 2000-01-01

Volume: 38

Issue: 4

Page Range: 299-306

Description:

The Muskingum-Cunge scheme applied to the one-dimensional unsteady advection-diffusion equation is investigated. To eliminate the numerical diffusion, the coefficients of the scheme are defined in such a way that the scheme does not contain the weighting parameters explicitly, but the Courant and Péclet numbers only. If one of the weighting factors is prescribed, the other should be necessarily negative in a lot of cases, which does not affect the applicability of the scheme. It is shown that the accuracy can be increased further, the numerical oscillations can also be eliminated by prescribing a simple relationship between the Courant and Péclet numbers. Sufficient conditions for strong stability are also presented.

Open Access: Yes

DOI: 10.1080/00221680009498329

Multigrid technique for biharmonic interpolation with application to dual and multiple reciprocity method

Csaba Gáspár

Publication Name: Numerical Algorithms

Publication Date: 1999-01-01

Volume: 21

Issue: 1-4

Page Range: 165-183

Description:

A biharmonic-type interpolation method is presented to solve 2D and 3D scattered data interpolation problems. Unlike the methods based on radial basis functions, which produce a large linear system of equations with fully populated and often non-selfadjoint and ill-conditioned matrix, the presented method converts the interpolation problem to the solution of the biharmonic equation supplied with some non-usual boundary conditions at the interpolation points. To solve the biharmonic equation, fast multigrid techniques can be applied which are based on a non-uniform, non-equidistant but Cartesian grid generated by the quadtree/octtree algorithm. The biharmonic interpolation technique is applied to the multiple and dual reciprocity method of the BEM to convert domain integrals to the boundary. This makes it possible to significantly reduce the computational cost of the evaluation of the appearing domain integrals as well as the memory requirement of the procedure. The resulting method can be considered as a special grid-free technique, since it requires no domain discretisation.

Open Access: Yes

DOI: 10.1023/a:1019173816678

A multipole expansion technique in solving boundary integral equations

Csaba Gáspár

Publication Name: Computer Methods in Applied Mechanics and Engineering

Publication Date: 1998-05-11

Volume: 157

Issue: 3-4

Page Range: 289-297

Description:

The Boundary Integral Equation Method reduces the spatial dimension of an elliptic problem by converting the original n-dimensional partial differential equation to an (n - 1)-dimensional boundary integral equation defined on the boundary of the domain of the original problem. At the same time, the discretisation of the problem is also remarkably simplified. The price of these advantages, however, is that the structure, as well as the algebraic properties of the resulting boundary element matrices, are somewhat unpleasant, since they are neither sparse nor self-adjoint in general, even if the original problem is self-adjoint. Consequently, the computational cost of the Boundary Integral Equation Method seems to be unnecessarily high. To make the method more economic from a computational point of view, we present a numerical technique based on the multipole expansion method, which reduces the computational cost of the appearing matrix-vector multiplications (i.e. the evaluations of the discretized boundary integrals) by a remarkable amount. The method is applicable also to die reconstruction problem, when the inner solution is to be reconstructed from the boundary solutions. © 1998 Elsevier Science S.A.

Open Access: Yes

DOI: 10.1016/S0045-7825(97)00241-7