Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains

An, Jing; Li, Huiyuan; Zhang, Zhimin

doi:10.1007/s11075-019-00760-4

Cited by 11 publications

(4 citation statements)

References 25 publications

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“…But this method has strict requirements for the calculation region, usually product type regions such as rectangles. In recent years, a sequence of efficient spectral methods for elliptic problems have been proposed, for example, [10] and [17] presented a dimension reduction spectral method for fourth-order problems in a circular and spherical domain, respectively, [8] proposed an efficient spectral approximation method for fourth-order Steklov problems in a circle, and [1] gave a spectral approximation and error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains. To our knowledge, there are few reports on the spectral method of fourth order equations in cylindrical regions because of the singularities and variable coefficients introduced by the cylindrical coordinate transformation, and in this case, both error analysis and algorithm implementation are challenging.…”

Section: Introductionmentioning

confidence: 99%

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A Legendre spectral‐Galerkin method for fourth‐order problems in cylindrical regions

Zheng,

2023

Numerical Methods Partial

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A spectral‐Galerkin method based on Legendre‐Fourier approximation for fourth‐order problems in cylindrical regions is studied in this paper. By the cylindrical coordinate transformation, a three‐dimensional fourth‐order problem in a cylindrical region is transformed into a sequence of decoupled fourth‐order problems with two dimensions and the corresponding pole conditions are also derived. With appropriately constructed weighted Sobolev space, a weak form is established. Based on this weak form, a spectral‐Galerkin discretization scheme is proposed and its error is rigorously analyzed by defining a new class of projection operators. Then, a set of efficient basis functions are used to write the discrete scheme as the linear systems with a sparse matrix based on tensor product. Numerical examples are presented to show the efficiency and high‐accuracy of the developed method. Finally, an application of the proposed method to the fourth‐order Steklov problem and the corresponding numerical experiments once again confirm the efficiency and spectral accuracy of the method.

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Section: Introductionmentioning

confidence: 99%

“…FIGURE1 The exact solution w(x) (left) and its numerical solution w MN (x) (right) when N = 35 and M = 15.…”

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A Legendre spectral‐Galerkin method for fourth‐order problems in cylindrical regions

Zheng,

2023

Numerical Methods Partial

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“…Thus it will cost expensive time and memory to perform in the computer. (2) Polar coordinates transformation will introduce pole singularity, which brings challenges to the theoretical analysis and the design of the algorithm.…”

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confidence: 99%

“…According to the references [2,3], the essential pole conditions need to be introduced to eliminate the pole singularity, which is caused from the transformation of polar coordinate. That is, φm (r) needs to satisfy the following pole conditions:…”

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A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

Jiang

Zhou

2023

DCDS-B

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<p style='text-indent:20px;'>Based on high order polynomial approximation and dimension reduction technique, we propose a novel numerical method for the fourth order Steklov problems in the circular domain. We first decompose the primal problem into a set of 1D problems via polar coordinate transformation and Fourier basis functions expansion. Then, by introducing a non-uniformly weighed Sobolev space, the variational form and corresponding discrete scheme are derived. Employing the Lax-Milgram lemma and approximation properties of the projection operators, we further prove existence and uniqueness of weak solutions and approximation solutions for each one-dimensional problems, and the error estimation between them, respectively. We also carry out ample numerical experiments which illustrate that the numerical algorithm is efficient and highly accurate.</p>

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Spectral Galerkin Approximation and Error Analysis Based on a Mixed Scheme for Fourth‐Order Problems in Complex Regions

Zheng,

2024

Numerical Methods Partial

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In this article, an efficient spectral Galerkin method, which is based on a mixed scheme, is proposed and studied for solving fourth‐order problems in complex regions. The fundamental idea behind this approach is to transform the initial problem into an equivalent form in cylindrical coordinates and to reshape the computational domain into a product‐type rectangular one, which facilitates the utilization of spectral methods. However, when considering the equivalent fourth‐order form directly in cylindrical coordinates, it introduces intricate pole conditions and variable coefficients, posing challenges to both theoretical analysis and algorithm implementation. To address this, we employ the orthogonality of Fourier series to further decompose it into a sequence of decoupled two‐dimensional fourth‐order eigenvalue problems. For each such problem, we introduce an auxiliary function to transform it into an equivalent second‐order coupled system. Building on this, we formulate a mixed variational formulation and discrete scheme, and prove the error estimates for eigenvalue and eigenfunction approximations. Furthermore, we extend this algorithm to the two‐dimensional complex domains. Finally, a series of numerical examples are presented, and the numerical results validate the effectiveness of the algorithm and the correctness of the theoretical results.

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Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains

Cited by 11 publications

References 25 publications

A Legendre spectral‐Galerkin method for fourth‐order problems in cylindrical regions

A Legendre spectral‐Galerkin method for fourth‐order problems in cylindrical regions

A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

Spectral Galerkin Approximation and Error Analysis Based on a Mixed Scheme for Fourth‐Order Problems in Complex Regions

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