26 results on '"Biharmonic equation"'
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2. A new mixed finite-element method for H2 elliptic problems.
- Author
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Farrell, Patrick E., Hamdan, Abdalaziz, and MacLachlan, Scott P.
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FINITE element method , *DIFFERENTIAL equations , *MULTIGRID methods (Numerical analysis) , *LINEAR systems , *BIHARMONIC equations , *LAGRANGE multiplier - Abstract
Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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3. A C0-conforming DG finite element method for biharmonic equations on triangle/tetrahedron.
- Author
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Ye, Xiu and Zhang, Shangyou
- Subjects
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FINITE element method , *YANG-Baxter equation , *TETRAHEDRA , *BIHARMONIC equations , *TRIANGLES , *VECTOR valued functions - Abstract
A C0-conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H2-norm and the L2-norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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4. A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes.
- Author
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Ye, Xiu and Zhang, Shangyou
- Subjects
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BIHARMONIC equations , *FINITE element method , *CONTINUOUS functions - Abstract
A conforming discontinuous Galerkin (CDG) C 0 - P k finite element method is introduced for solving the biharmonic equation on triangular and tetrahedral meshes. A C 0 - P k finite element function is a continuous and piecewise polynomial of degree k on a triangular or tetrahedral mesh. The CDG method is based on taking weak divergence on the gradient of C 0 - P k finite elements. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established. Numerical results are presented to verify the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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5. High order Morley elements for biharmonic equations on polytopal partitions.
- Author
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Li, Dan, Wang, Chunmei, Wang, Junping, and Zhang, Shangyou
- Subjects
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BIHARMONIC equations , *FINITE element method , *SCHUR complement , *GALERKIN methods , *DEGREES of freedom - Abstract
This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H 2 norm and the usual L 2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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6. Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes.
- Author
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Ye, Xiu and Zhang, Shangyou
- Subjects
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FINITE element method , *BIHARMONIC equations , *DISCONTINUOUS functions , *BILINEAR forms , *GALERKIN methods - Abstract
In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous P k polynomials are employed. To connect discontinuous functions, the inter-element traces, { u h } and { ∇ u h } , are usually defined as some averages in discontinuous Galerkin finite element methods. But in this CDG finite element method, they are defined as projections of a lifted P k + 4 polynomial from four P k polynomials on neighboring triangles. With properly chosen weak Hessian spaces, when tested by discontinuous polynomials, the variation form can have no inter-element integral, neither any stabilizer. That is, the bilinear form is the same as that of conforming finite elements for solving the biharmonic equation. Such a conforming discontinuous Galerkin finite element method converges four orders above the optimal order, i.e., the P k solution has an O (h k + 5) convergence in L 2 -norm, and an O (h k + 3) convergence in H 2 -norm. A local post-process is defined, which lifts the P k solution to a P k + 4 quasi-optimal solution. Numerical tests are provided, confirming the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. Pointwise error estimates for C0 interior penalty approximation of biharmonic problems.
- Author
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Leykekhman, D.
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GREEN'S functions , *BIHARMONIC equations , *APPROXIMATION error , *ESTIMATES - Abstract
The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the C0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green's function estimates for the continuous problem which has independent interest. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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8. Generalized weak Galerkin finite element methods for biharmonic equations.
- Author
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Li, Dan, Wang, Chunmei, and Wang, Junping
- Subjects
- *
FINITE element method , *BIHARMONIC equations - Abstract
The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H 2 norm and a L 2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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9. EFFICIENT BLOCK PRECONDITIONING FOR A Cҡ FINITE ELEMENT DISCRETIZATION OF THE DIRICHLET BIHARMONIC PROBLEM.
- Author
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PESTANA, J., MUDDLE, R., HEIL, M., TISSEUR, F., and MIHAJLOVIĆ, M.
- Subjects
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BIHARMONIC equations , *NUMERICAL solutions to differential equations , *NUMERICAL analysis , *FINITE element method , *CONJUGATE gradient methods , *ALGEBRAIC multigrid methods - Abstract
We present an efficient block preconditioner for the two-dimensional biharmonic Dirichlet problem discretized by C¹ bicubic Hermite finite elements. In this formulation each node in the mesh has four different degrees of freedom (DOFs). Grouping DOFs of the same type together leads to a natural blocking of the Galerkin coefficient matrix. Based on this block structure, we develop two preconditioners: a 2 x 2 block diagonal (BD) preconditioner and a block bordered diagonal (BBD) preconditioner. We prove mesh-independent bounds for the spectra of the BDpreconditioned Galerkin matrix under certain conditions. The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite (SPD) and assembled from element matrices. We demonstrate the effectiveness of an inexact version of the BBD preconditioner, which exhibits near-optimal scaling in terms of computational cost with respect to the discrete problem size. Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion, and nonconvex domains. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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10. Interpolating minimal energy C1-Surfaces on Powell- Sabin Triangulations: Application to the resolution of elliptic problems.
- Author
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Fortes, M. A., González, P., Ibáñez, M. J., and Pasadas, M.
- Subjects
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FINITE element method , *TECHNOLOGY convergence , *PARTIAL differential equations , *DIFFERENTIAL equations , *NUMERICAL analysis - Abstract
In this article, we present a method to obtain a C1-surface, defined on a bounded polygonal domain Ω, which interpolates a specific dataset and minimizes a certain 'energy functional.' The minimization space chosen is the one associated to the Powell-Sabin finite element, whose elements are C1-quadratic splines. We develop a general theoretical framework for that, and we consider two main applications of the theory. For both of them, we give convergence results, and we present some numerical and graphical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 798-821, 2015 [ABSTRACT FROM AUTHOR]
- Published
- 2015
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11. Solving biharmonic equation using the localized method of approximate particular solutions.
- Author
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Li, Ming, Amazzar, Ghizlane, Naji, Ahmed, and Chen, C.S.
- Subjects
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BIHARMONIC equations , *APPROXIMATE solutions (Logic) , *FINITE element method , *FINITE difference method , *POISSON processes , *NUMERICAL solutions to differential equations , *RADIAL basis functions - Abstract
Some localized numerical methods, such as finite element and finite difference methods (FDMs), have encountered difficulties when solving fourth or higher order differential equations. Localized methods, which use radial basis functions, are considered the generalized FDMs and, thus, inherit the similar difficulties when solving higher order differential equations. In this paper, we deal with the use of the localized method of approximate particular solutions (LMAPS), a recently developed localized radial basis function collocation method, in solving two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmonic problem into two Poisson equations, and then the LMAPS is applied to each Poisson's problem to compute numerical solutions. Furthermore, the influence of the shape parameter and different radial basis functions on the numerical solution is discussed. The effectiveness of the proposed method is demonstrated by solving three examples in both regular and irregular domains. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
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12. A finite element method for a biharmonic equation based on gradient recovery operators.
- Author
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Lamichhane, Bishnu
- Subjects
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FINITE element method , *APPROXIMATION theory , *BIHARMONIC equations , *SAMPLING errors , *BIORTHOGONAL systems - Abstract
A new non-conforming finite element method is proposed for the approximation of the biharmonic equation with clamped boundary condition. The new formulation is based on a gradient recovery operator. Optimal a priori error estimates are proved for the proposed approach. The approach is also extended to cover a singularly perturbed problem. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
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13. A comparison of a posteriori error estimates for biharmonic problems solved by the FEM
- Author
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Segeth, Karel
- Subjects
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COMPARATIVE studies , *ERROR analysis in mathematics , *BIHARMONIC equations , *FINITE element method , *NUMERICAL analysis , *PARTIAL differential equations , *MATHEMATICAL analysis - Abstract
Abstract: The classical a posteriori error estimates are mostly oriented to the use in the finite element -methods while the contemporary higher-order -methods usually require new approaches in a posteriori error estimation. These methods hold a very important position among adaptive numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications. In the paper, we are concerned with a review and comparison of error estimation procedures for the biharmonic and some more general fourth order partial differential problems with special regards to the needs of the -method. We point out some advantages and drawbacks of analytical and computational a posteriori error estimates. [Copyright &y& Elsevier]
- Published
- 2012
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14. Fluid-structure interactions using different mesh motion techniques
- Author
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Wick, Thomas
- Subjects
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FLUID-structure interaction , *FINITE element method , *JACOBIAN matrices , *SMOOTHNESS of functions , *NONLINEAR systems , *GALERKIN methods , *LAGRANGIAN functions , *PARTIAL differential equations - Abstract
Abstract: In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structure interactions in arbitrary Lagrangian–Eulerian coordinates. The mesh movement is realized by solving an additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examine an implementation of time discretization that is designed with finite differences. Spatial discretization is based on a Galerkin finite element method. To solve the resulting discrete nonlinear systems, a Newton method with exact Jacobian matrix is used. Our results show that the biharmonic model produces the smoothest meshes but has increased computational cost compared to the other two approaches. [Copyright &y& Elsevier]
- Published
- 2011
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15. THE LOWEST ORDER DIFFERENTIABLE FINITE ELEMENT ON RECTANGULAR GRIDS.
- Author
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JUN HU, YUNQING HUANG, and SHANGYOU ZHANG
- Subjects
- *
FINITE element method , *NUMERICAL analysis , *BIQUADRATIC equations , *TENSOR products , *LINEAR algebra - Abstract
A macro type of biquadratic C¹ finite elements is constructed on rectangle grids. This is a rectangular version of the C¹ Powell-Sabin element, a C¹-P2 element on triangular grids. Here, each rectangle of the base grid is refined into four subrectangles. As in the case of the Powell-Sabin element, we have more constraints than the number of degrees of freedom on each macroelement. However, the extra constraints are consistent. It is shown further that the constructed finite element space is the full C¹-Q2 space on the grid. It is also shown that the finite element space is a tensor product space of one-dimensional C¹-P2 spaces, where the nodal basis is supported on four intervals. The B-spline function of P2 is supported on three intervals. The Girault-Scott operator is extended to the element. The application and the convergence of the finite element to the biharmonic equation are presented. Numerical tests are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
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16. RIESZ BASES OF WAVELETS AND APPLICATIONS TO NUMERICAL SOLUTIONS OF ELLIPTIC EQUATIONS.
- Author
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RONG-QING JIA and WEI ZHAO
- Subjects
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NUMERICAL solutions to elliptic equations , *RIESZ spaces , *SOBOLEV spaces , *WAVELETS (Mathematics) , *SPLINES , *APPROXIMATION theory , *FINITE element method , *BIHARMONIC equations - Abstract
We investigate Riesz bases of wavelets in Sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth-order. First, we study bicubic splines on the unit square with homogeneous boundary conditions. The approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation. Second, we develop a fairly general theory for Riesz bases of Hilbert spaces equipped with induced norms. Under the guidance of the general theory, we are able to construct wavelet bases for Sobolev spaces on the unit square. The condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded. Third, we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient. Finally, we extend our study to general elliptic equations of fourth-order and demonstrate that our numerical schemes also have superb performance in the general case. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
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17. A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems
- Author
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Lamichhane, Bishnu P.
- Subjects
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FINITE element method , *BIHARMONIC equations , *BIORTHOGONAL systems , *APPROXIMATION theory , *BOUNDARY value problems , *LAGRANGE equations , *MULTIPLIERS (Mathematical analysis) , *STOCHASTIC convergence - Abstract
Abstract: We propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a Lagrange multiplier as new unknowns. Working with a pair of bases forming a biorthogonal system, we can easily eliminate the gradient of the solution and the Lagrange multiplier from the saddle point system leading to a positive definite formulation. Using a superconvergence property of a gradient recovery operator, we prove an optimal a priori estimate for the finite element discretization for a class of meshes. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
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18. Powell–Sabin spline based multilevel preconditioners for the biharmonic equation
- Author
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Maes, Jan and Bultheel, Adhemar
- Subjects
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SPLINE theory , *BIHARMONIC equations , *FINITE element method , *ASYMPTOTIC expansions , *COMPUTATIONAL complexity , *NUMERICAL analysis - Abstract
Abstract: The Powell–Sabin (PS) piecewise quadratic finite element on the PS 12-split of a triangulation is a common choice for the construction of a BPX-type preconditioner for the biharmonic equation. In this note we investigate the related Powell–Sabin element on the PS 6-split instead of the PS 12-split for the construction of such preconditioners. For the PS 6-split element multilevel spaces can be created using a -refinement scheme instead of the traditional dyadic scheme. Topologically -refinement has many advantages: it is a slower refinement than the dyadic split operation, and it alternates the orientation of the refined triangles. Therefore we expect a reduction of the amount of work when compared to the PS 12-split element BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement. [Copyright &y& Elsevier]
- Published
- 2010
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19. Effective condition number of the Hermite finite element methods for biharmonic equations
- Author
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Li, Zi Cai, Huang, Hung-Tsai, and Huang, Jin
- Subjects
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FINITE element method , *BIHARMONIC equations , *PARTIAL differential equations , *MATHEMATICAL models , *EIGENVALUES , *FINITE differences - Abstract
Abstract: For biharmonic equations, the Hermite finite element methods (FEM) are chosen, to seek their approximate solutions. The linear algebraic equations are obtained from the Hermite FEM, where the matrix A is symmetric and positive definite, and x and b are the unknown and known vectors, respectively. It is well known that , and and are the maximal and minimal eigenvalues of the stiffness matrix A, respectively. The bounds of Cond are derived to be . Note that when h is small, the values of Cond () are huge, to indicate a severe instability, compared with Cond for Poisson''s equation by the FEM. In fact, for specific application problems, the instability is not so severe, a new effective condition number is defined by Cond_eff in [Z.C. Li, C.S. Chien, H.T. Huang, Effective condition number for finite difference method, Comput. Appl. Math. 198 (2007) 208–235], to provide a better upper bound of perturbation errors. It is proven that Cond_eff for general cases, which is smaller than the traditional Cond. However, for special cases, the Cond_eff could be much smaller. For instant, for the homogeneous boundary conditions of biharmonic equations, Cond_eff , can be reached as h diminishes. This is astonishing, against our intuition from the knowledge of the Cond. From the analysis in this paper, the traditional Cond may mislead the stability analysis for practical computation of engineering problems. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
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20. LOCAL A PRIORI AND A POSTERIORI ERROR ESTIMATE OF TQC9 ELEMENT FOR THE BIHARMONIC EQUATION.
- Author
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Ming Wang and Weimeng Zhang
- Subjects
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A priori , *BIHARMONIC equations , *FINITE element method , *ERRORS , *ESTIMATES - Abstract
In this paper, local a priori, local a posteriori and global a posteriori error estimates are obtained for TQC9 element for the biharmonic equation. An adaptive algorithm is given based on the a posteriori error estimates. [ABSTRACT FROM AUTHOR]
- Published
- 2008
21. hp-version interior penalty DGFEMs for the biharmonic equation
- Author
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Süli, Endre and Mozolevski, Igor
- Subjects
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FINITE element method , *GALERKIN methods , *BIHARMONIC equations , *A priori , *ERROR analysis in mathematics , *NUMERICAL analysis - Abstract
We construct hp-version interior penalty discontinuous Galerkin finite element methods (DGFEMs) for the biharmonic equation, including symmetric and nonsymmetric interior penalty discontinuous Galerkin methods and their combinations: semisymmetric methods. Our main concern is to establish the stability and to develop the a priori error analysis of these methods. We establish error bounds that are optimal in h and slightly suboptimal in p. The theoretical results are confirmed by numerical experiments. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
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22. Efficient parallel solvers for the biharmonic equation
- Author
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Mihajlović, Milan D. and Silvester, David J.
- Subjects
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LINEAR algebraic groups , *MULTIGRID methods (Numerical analysis) , *BIHARMONIC equations , *FINITE element method - Abstract
We examine the convergence characteristics and performance of parallelised Krylov subspace solvers applied to the linear algebraic systems that arise from low-order mixed finite element approximation of the biharmonic problem. Our strategy results in preconditioned systems that have nearly optimal eigenvalue distribution, which consists of a tightly clustered set together with a small number of outliers. We implement the preconditioner operator in a “black-box” fashion using publicly available parallelised sparse direct solvers and multigrid solvers for the discrete Dirichlet Laplacian. We present convergence and timing results that demonstrate efficiency and scalability of our strategy when implemented on contemporary computer architectures. [Copyright &y& Elsevier]
- Published
- 2004
- Full Text
- View/download PDF
23. Global superconvergence of finite element methods for biharmonic equations and blending surfaces
- Author
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Li, Z.C. and Lu, T.T.
- Subjects
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FINITE element method , *BIHARMONIC equations , *GEOMETRIC surfaces - Abstract
In this paper, new numerical algorithms of finite element methods (FEM) are reported for both biharmonic equations and 3D blending surfaces, to achieve the global superconvergence O(h3)-O(h4) in H2 norms. This is significant, compared with the optimal convergence O(h2). The algorithms are simple because only an a posteriori interpolant solution is needed. Such a global convergence method was originated by Lin and his colleagues in [1–3] for only the clamped boundary conditions. Recently, we extended the global superconvergence to other boundary conditions, such as the simple support condition, the periodic boundary, and the natural boundary condition. Moreover, we apply in [4–6] this global superconvergence to the FEM using the penalty techniques for biharmonic equations and blending problems, also to reach O(h3) and O(h4) for quasiuniform and uniform □ij, respectively. Currently, we develop in [7,8] and in this paper the FEM using the penalty plus hybrid techniques to reduce the condition number down to O(h−4)-O(h−5) of the associated matrix, while retaining superconvergence O(h3)-O(h4). Since instability is severe for biharmonic equations, any reduction of the condition number is crucial. By the new algorithms in this paper, not only can a great deal of CPU time be saved, but also the complicated biharmonic equations and blending surfaces may be solved in double precision. Numerical experiments are carried out to support the theoretical conclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2002
- Full Text
- View/download PDF
24. A LEAST SQUARES METHOD OR SOLVING BIHARMONIC PROBLEMS.
- Author
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Thatcher, R. W.
- Subjects
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EQUATIONS , *BIHARMONIC equations , *FINITE element method , *STOCHASTIC convergence , *CAUCHY-Riemann equations , *MATHEMATICAL functions - Abstract
We discuss a generalization of the CauchyRiemann equations, which is applicable to biharmonic problems. A first-order system is derived from these equations, and this system is solved using the least squares method with continuous trial functions set up by the finite element method. Optimal rates of convergence in L2(Ω) are proved in regions with smooth boundaries and optimal rates of convergence are proved in H1(Ω) and, using linear triangular elements, observed in L2(Ω) in polygonal regions. [ABSTRACT FROM AUTHOR]
- Published
- 2000
- Full Text
- View/download PDF
25. A SCALABLE SUBSTRUCTURING METHOD BY LAGRANGE MULTIPLIERS FOR PLATE BENDING PROBLEMS.
- Author
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Mandel, Jan, Tezaur, Radek, and Farhat, Charbel
- Subjects
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NUMERICAL solutions to Lagrange equations , *NUMERICAL analysis , *DIFFERENTIAL equations , *BESSEL functions , *CALCULUS , *FINITE element method , *MATHEMATICAL analysis - Abstract
We present a new Lagrange multiplier-based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the finite element tearing and interconnecting substructuring algorithm to the biharmonic equation. The main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations. The resulting method is proved to have a condition number that does not grow with the number of subdomains but rather grows at most polylogarithmically with the number of elements per subdomain. These optimal properties hold for numerous plate bending elements that are used in practice including the HsiehCloughTocher element, the discrete Kirchhoff triangle, and a class of nonlocking elements for the ReissnerMindlin plate models. Computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method. Computational efficiency is also demonstrated with the numerical solution in 45 iterations and 105 seconds on a 64-processor IBM SP2 of a plate bending problem with almost one million degrees of freedom. [ABSTRACT FROM AUTHOR]
- Published
- 1999
- Full Text
- View/download PDF
26. Multilevel Preconditioners for Discretizations of the Biharmonic Equation by Rectangular Finite Elements.
- Author
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Oswald, Peter
- Subjects
- *
FINITE element method , *NUMERICAL analysis , *BIHARMONIC equations , *PARTIAL differential equations , *SYSTEMS theory , *LINEAR systems - Abstract
Recently, some new multilevel preconditioners for solving elliptic finite element discretizations by iterative methods have been proposed. They are based on appropriate splittings of the finite element spaces under consideration, and may be analyzed within the framework of additive Schwarz schemes. In this paper we discuss some multilevel methods for discretizations of the fourth-order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner-Fox-Schmit rectangle, the generalization of the Bramble-Pasciak-Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived. [ABSTRACT FROM AUTHOR]
- Published
- 1995
- Full Text
- View/download PDF
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