A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations

2009 ◽  
Vol 25 (3) ◽  
pp. 712-739 ◽  
Author(s):  
Eid H. Doha ◽  
Ali H. Bhrawy
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Elliptic Differential Equations ◽  
Spectral Galerkin Method

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850045 ◽  
Author(s):  
B. V. Rathish Kumar ◽  
Gopal Priyadarshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Stability Estimates ◽  
Partial Differential ◽  
Wavelet Galerkin Method

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

scholarly journals TWO-GRID WEAK GALERKIN METHOD FOR SEMILINEAR ELLIPTIC DIFFERENTIAL EQUATIONS

2022 ◽  
Author(s):  
luoping chen ◽  
fanyun wu ◽  
guoyan zeng
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Mesh Size ◽  
Initial Guess ◽  
Coarse Mesh ◽  
Weak Galerkin ◽  
Fine Mesh ◽  
Semilinear Elliptic ◽  
Elliptic Differential Equations ◽  
Linearized System

In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semi-linear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as a initial guess to linearize the semilinear equation on the fine mesh, i.e., on the fine mesh (with mesh size $h$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and $h=H^2$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

scholarly journals Uniqueness Results for Higher Order Elliptic Equations in Weighted Sobolev Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Loredana Caso ◽  
Patrizia Di Gironimo ◽  
Sara Monsurrò ◽  
Maria Transirico
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Weighted Sobolev Spaces ◽  
Discontinuous Coefficients ◽  
Uniqueness Results ◽  
Elliptic Differential Equations

We prove some uniqueness results for the solution of two kinds of Dirichlet boundary value problems for second- and fourth-order linear elliptic differential equations with discontinuous coefficients in polyhedral angles, in weighted Sobolev spaces.

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