scholarly journals Riesz bases of wavelets and applications to numerical solutions of elliptic equations

2011 ◽  
Vol 80 (275) ◽  
pp. 1525-1556 ◽  
Author(s):  
Rong-Qing Jia ◽  
Wei Zhao
Keyword(s):  
Elliptic Equations ◽  
Numerical Solutions ◽  
Riesz Bases

scholarly journals Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1381
Author(s):  
Jinru Wang ◽  
Wenhui Shi ◽  
Lin Hu
Keyword(s):  
Elliptic Equations ◽  
Numerical Solutions ◽  
Homogeneous Boundary Condition ◽  
Riesz Bases ◽  
Condition Numbers ◽  
Order Elliptic Equation ◽  
Approximation Properties ◽  
Numerical Examples ◽  
Stiffness Matrices ◽  
Second Order Elliptic Equation

This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [0,1] and consider their approximation properties. Then we define the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efficiently.

DISCRETE CONSERVATION AND DISCRETE MAXIMUM PRINCIPLE FOR ELLIPTIC PDEs

1998 ◽  
Vol 08 (04) ◽  
pp. 685-711 ◽  
Author(s):  
ENRICO BERTOLAZZI
Keyword(s):  
Maximum Principle ◽  
Differential Operator ◽  
Finite Volume ◽  
Elliptic Equations ◽  
Numerical Solutions ◽  
Strong Maximum Principle ◽  
Discrete Maximum Principle ◽  
Elliptic Pdes ◽  
Numerical Schemes

A class of finite volume numerical schemes for the solution of self-adjoint elliptic equations is described. The main feature of the schemes is that numerical solutions share both discrete conservation and discrete strong maximum principle without restriction on the differential operator or on the volume elements.

A refined convergence analysis of multigrid algorithms for elliptic equations

2015 ◽  
Vol 13 (03) ◽  
pp. 255-290 ◽  
Author(s):  
Qianshun Chang ◽  
Rong-Qing Jia
Keyword(s):  
General Theory ◽  
Convergence Analysis ◽  
Elliptic Equations ◽  
Numerical Solutions ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Multigrid Algorithm ◽  
Seidel Method ◽  
Wide Range ◽  
Additional Regularity

Multigrid algorithms, in particular, multigrid V-cycles, are investigated in this paper. We establish a general theory for convergence of the multigrid algorithm under certain approximation conditions and smoothing conditions. Our smoothing conditions are satisfied by commonly used smoothing operators including the standard Gauss–Seidel method. Our approximation conditions are verified for finite element approximation to numerical solutions of elliptic partial differential equations without any requirement of additional regularity of the solution. Our convergence analysis of multigrid algorithms can be applied to a wide range of problems. Numerical examples are also provided to illustrate the general theory.

scholarly journals Numerical solution of nonlinear elliptic systems by block monotone iterations

2019 ◽  
Vol 60 ◽  
pp. C79-C94
Author(s):  
Mohamed Saleh Mehdi Al-Sultani ◽  
Igor Boglaev
Keyword(s):  
Iterative Methods ◽  
Elliptic Equations ◽  
Numerical Solutions ◽  
Upper And Lower Solutions ◽  
Elliptic Systems ◽  
Nonlinear Elliptic Equations ◽  
Difference Schemes ◽  
Uniqueness Of Solutions ◽  
Monotone Iterative ◽  
Nonlinear Elliptic

We present numerical methods for solving a coupled system of nonlinear elliptic problems, where reaction functions are quasimonotone nondecreasing. We utilize block monotone iterative methods based on the Jacobi and Gauss--Seidel methods incorporated with the upper and lower solutions method. A convergence analysis and the theorem on uniqueness of solutions are discussed. Numerical experiments are presented. References Boglaev, I., Monotone iterates for solving systems of semilinear elliptic equations and applications, ANZIAM J, Proceedings of the 8th Biennial Engineering Mathematics and Applications Conference, EMAC-2007, 49(2008), C591C608. doi:10.21914/anziamj.v49i0.311 Pao, C. V., Nonlinear parabolic and elliptic equations, Springer-Verlag (1992). doi:10.1007/978-1-4615-3034-3 Pao, C. V., Block monotone iterative methods for numerical solutions of nonlinear elliptic equations, Numer. Math., 72(1995), 239262. doi:10.1007/s002110050168 Samarskii, A., The theory of difference schemes, CRC Press (2001). https://www.crcpress.com/The-Theory-of-Difference-Schemes/Samarskii/p/book/9780824704681 Varga, R. S., Matrix iterative analysis, Springer-Verlag (2000). doi:10.1007/978-3-642-05156-2

scholarly journals Numerical approximations for fractional elliptic equations via the method of semigroups

2020 ◽  
Vol 54 (3) ◽  
pp. 751-774
Author(s):  
Nicole Cusimano ◽  
Félix del Teso ◽  
Luca Gerardo-Giorda
Keyword(s):  
Boundary Conditions ◽  
Singular Integral ◽  
Elliptic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Heat Semigroup ◽  
Fractional Sobolev Space ◽  
Novel Approach ◽  
The Family ◽  
Element Discretizations

We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations (−Δ)su = f in Ω, subject to some homogeneous boundary conditions B on ∂Ω, where s ∈ (0,1), Ω ⊂ ℝn is a bounded domain, and (-Δ)s is the spectral fractional Laplacian associated to B on ∂Ω. We use the solution representation (−Δ)−s f together with its singular integral expression given by the method of semigroups. By combining finite element discretizations for the heat semigroup with monotone quadratures for the singular integral we obtain accurate numerical solutions. Roughly speaking, given a datum f in a suitable fractional Sobolev space of order r ≥ 0 and the discretization parameter h > 0, our numerical scheme converges as O(hr+2s), providing super quadratic convergence rates up to O(h4) for sufficiently regular data, or simply O(h2s) for merely f ∈ L2 (Ω). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.

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