System Of Elliptic Partial Differential Equations Research Articles

Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.

Localized boundary-domain singular integral equations of the Robin type problem for self-adjoint second-order strongly elliptic PDE systems

Abstract The paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under consideration is reduced to the a system of localized boundary-domain singular integral equations (LBDSIE). The equivalence between the original boundary value problem and the corresponding LBDSIE system is established. The matrix operator generated by the LBDSIE system belongs to the Boutet de Monvel algebra. With the help of the Vishik–Eskin theory based on the Wiener–Hopf factorization method, the Fredholm properties of the corresponding localized boundary-domain singular integral operator are investigated and its invertibility in appropriate function spaces is proved.

Deep least-squares methods: An unsupervised learning-based numerical method for solving elliptic PDEs

This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine parameters of the deep neural network. There are various least-squares functionals for a partial differential equation. This paper focuses on the so-called first-order system least-squares (FOSLS) functional studied in [3], which is based on a first-order system of scalar second-order elliptic PDEs. Numerical results for second-order elliptic PDEs in one dimension are presented.

Optimized Schwarz Methods for the Optimal Control of Systems Governed by Elliptic Partial Differential Equations

Optimal control of systems governed by elliptic partial differential equations (PDEs) without constraint on the set of controls can be equivalently reformulated as a coupled system of second order elliptic PDEs, which has been considered to solve by a non-overlapping Schwarz domain decomposition method with the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions by Benamou (SIAM J Numer Anal 33:2401–2416, 1996) where a convergence analysis had been performed. Towards fast convergence of the overlapping and non-overlapping Schwarz subdomain iterations, in this paper we firstly perform, for fixed Tikhonov parameter $$\mu $$ , rigorous analyses based on optimization of the convergence factor of subdomain iterations in Fourier frequency domain to give the optimized transmission parameters involved in the above mentioned transmission conditions, as well as those involved in a Ventcell-like and a two-sided Robin-like transmission condition that we propose to accelerate the Schwarz subdomain iterations, and meanwhile we obtain also the corresponding asymptotic convergence rate estimates. The results show that the Tikhonov parameter $$\mu $$ occurs in both the optimized transmission parameters and the corresponding convergence rate estimates and affects the performance of the Schwarz domain decomposition methods significantly: when $$\mu $$ is less than a certain threshold value, with the decreasing of the Tikhonov parameter $$\mu $$ , the subdomain iteration converges more and more fast, though the regularity of the system deteriorates in this process. We lastly investigate the case where the Tikhonov parameter $$\mu =h^4$$ that is suggested by Benamou (where h is the mesh size). We obtain as well the optimized transmission parameters involved in the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions, and find that they lead to optimized Schwarz methods that are very robust in the mesh size. The analysis also sheds light on optimizing the Schwarz domain decomposition methods for biharmonic equations, since they can also be reformulated as a system of second order elliptic PDEs. We use various numerical experiments to illustrate the theoretical findings.

Fast Fourier transform solvers and preconditioners for quadratic spline collocation

Quadratic Spline Collocation (QSC) methods of optimal order of convergence have been recently developed for the solution of elliptic Partial Differential Equations (PDEs). In this paper, linear solvers based on Fast Fourier Transforms (FFT)are developed for the solution of the QSC equations. The complexity of the FFT solvers is O(N 2 log N), where N is the gridsize in one dimension. These direct solvers can handle PDEs with coefficients in one variable or constant, and Dirichlet, Neumann, alternating Dirichlet-Neumann or periodic boundary conditions, along at least one direction of a rectangular domain. General variable coefficient PDEs are handled by preconditioned iterative solvers. The preconditioner is the QSC matrix arising from a constant coefficient PDE. The convergence analysis of the preconditioner is presented. It is shown that, under certain conditions, the convergence rate is independent of the gridsize. The preconditioner is solved by FFT techniques, and integrated with one-step or acceleration methods, giving rise to asymptotically almost optimal linear solvers, with complexity O(N 2 log N). Numerical experiments verify the effectiveness of the solvers and preconditioners, even on problems more general than the analysis assumes. The development and analysis of FFT solvers and preconditioners is extended to QSC equations corresponding to systems of elliptic PDEs.