High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral methods. These tools make it difficult to apply these techniques to higher dimensions. In contrast, fixed Cartesian grid methods, such as the immersed boundary (IB) method, are easy to apply and generalize, but typically are low-order accurate. In this study, we introduce the Smooth Forcing Extension (SFE) method, a fixed Cartesian grid technique that builds on the insights of the IB method, and allows one to obtain arbitrary orders of accuracy. Our approach relies on a novel Fourier continuation method to compute extensions of the inhomogeneous terms to any desired regularity. This is combined with the highly accurate Non-Uniform Fast Fourier Transform for interpolation operations to yield a fast and robust method. Numerical tests confirm that the technique performs precisely as expected on one-dimensional test problems. In higher dimensions, the performance is even better, in some cases yielding sub-geometric convergence. We also demonstrate how this technique can be applied to solving parabolic problems and for computing the eigenvalues of elliptic operators on general domains, in the process illustrating its stability and amenability to generalization.
We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is quasi-optimal in two dimensional space and sub-optimal in three dimensional space. Numerical simulations are provided to confirm our findings.
We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The method we propose is capable of dealing with either variational and non-variational problems, and because of its meshless nature, it can also deal with problems posed in high-dimensional domains. We prove the $Γ$-convergence of the neural network approximation towards the solution of the continuous problem, and extend the convergence proof to some well-known related methods. Finally, we present several numerical examples illustrating the performance of our…
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adam…
We investigate the Cauchy problem for elliptic operators with $C^\infty$-coefficients at a regular set $Ω\subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the subset $Γ\subset \partialΩ$ and our objective is to reconstruct the trace of the $H^1(Ω)$ solution of an elliptic equation at $\partial Ω/ Γ$. The method described here is a generalization of the algorithm developed by Maz'ya et al. [Ma] for the Laplace operator, who proposed a method based on solving successive well-posed mixed boundary value problems (BVP) using the given Cauchy data as pa…
While deep learning has achieved remarkable success in solving partial differential equations (PDEs), it still faces significant challenges, particularly when the PDE solutions have low regularity or singularities. To address these issues, we propose the Weak TransNet (WTN) method, based on a Petrov-Galerkin formulation, for solving elliptic PDEs in this work, though its framework may extend to other classes of equations. Specifically, the neural feature space defined by TransNet (Zhang et al., 2023) is used as the trial space, while the test space is composed of radial basis functions. Since …
We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of t…
An economic cascadic tensor multigrid method for solving high dimensional elliptic linear partial differential problems
Guilin University of Electronic Technology（桂林电子科技大学）；Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation（广西高校数据分析与计算重点实验室）；Center for Applied Mathematics of Guangxi (GUET)（广西应用数学中心）
In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM), and its local kernel variants, to approximate second-order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points at the boundary. The resulting GPDM estimator restr…
In this paper we consider one particular mathematical problem of this large area of fractional powers of self-adjoined elliptic operators, defined either by Dunford-Taylor-like integrals or by the representation through the spectrum of the elliptic operator. Due to the mathematical modeling of various non-local phenomena using such operators recently a number of numerical methods for solving equations involving operators of fractional order were introduced, studied, and tested. Here we consider the discrete counterpart of such problems obtained from finite difference or finite element approxim…
In this article, we develop and analyze a finite element method with the first family Nédélec elements of the lowest degree for solving a Maxwell interface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation on unfitted meshes. To capture the jump conditions optimally, we construct and use $\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interface elements while keep using the standard Nédélec functions on all the non-interface elements. We establish a few important properties for the IFE functions including the unisolvence according to the edge degrees of free…
We consider the Vlasov-Fokker-Planck equation with random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with suitable assumption on the anisotropy of the randomness, adopting the technique employed in elliptic PDEs [Cohen, DeVore, 2015], we prove the best N approximation in the random space breaks the dimension curse and the convergence rate is faster than the Monte Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in [Chkifa, Co…
We present an efficient algorithm to solve elliptic Dirichlet problems defined on the cluster of $\mathbb{Z}^d$ supercritical Bernoulli percolation, as a generalization of the iterative method proposed by S. Armstrong, A. Hannukainen, T. Kuusi and J.-C. Mourrat. We also explore the two-scale expansion on the infinite cluster of percolation, and use it to give a rigorous analysis of the algorithm.
The key innovation in this paper is an open-source, high-performance iterative solver for high contrast, strongly anisotropic elliptic partial differential equations implemented within dune-pdelab. The iterative solver exploits a robust, scalable two-level additive Schwarz preconditioner, GenEO (Spillane et al. 2014). The development of this solver has been motivated by the need to overcome the limitations of commercially available modeling tools for solving structural analysis simulations in aerospace composite applications. Our software toolbox dune-composites encapsulates the mathematical c…
We present a new numerical method for solving the elliptic homogenization problem. The main idea is that the missing effective matrix is reconstructed by solving the local least-squares in an offline stage, which shall be served as the input data for the online computation. The accuracy of the proposed method are analyzed with the aid of the refined estimates of the reconstruction operator. Two dimensional and three dimensional numerical tests confirm the efficiency of the proposed method, and illustrate that this online-offline strategy may significantly reduce the cost without loss of accura…
In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, ``[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless march to larger and larger [bandwidths] continues''. This paper attempts to further the virtue of the Fast Fourier Transform (FFT) as not only bandwidth is pushed to its limits, but also the dimension of the problem. Instead of using the traditional FFT however, we make a key substitution: a high-dimensional, sparse Fourier transform (SFT) paired with rando…
The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for solutions based on samples of these Markov processes which have advantages over traditional numerical methods in some cases. However, naïve numerical implementations suffer from statistical bias and sampling error. We present methods to discretize the stochastic process appearing in the Feynman-Kac formula that reduce the bias of the numerical scheme. We also propo…
The Scheduled Relaxation Jacobi (SRJ) method is a linear solver algorithm which greatly improves the convergence of the Jacobi iteration through the use of judiciously chosen relaxation factors (an SRJ scheme) which attenuate the solution error. Until now, the method has primarily been used to accelerate the solution of elliptic PDEs (e.g. Laplace, Poisson's equation) as the currently available schemes are restricted to solving this class of problems. The goal of this paper is to present a methodology for constructing SRJ schemes which are suitable for solving non-elliptic PDEs (or equivalent,…
This paper introduces a deep learning method for solving an elliptic hemivariational inequality (HVI). In this method, an expectation minimization problem is first formulated based on the variational principle of underlying HVI, which is solved by stochastic optimization algorithms using three different training strategies for updating network parameters. The method is applied to solve two practical problems in contact mechanics, one of which is a frictional bilateral contact problem and the other of which is a frictionless normal compliance contact problem. Numerical results show that the dee…
We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation to the gradient in a piecewisely irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in continuous finite element space. The variational setting naturally provides a posteriori error which could be used in an adaptive refinement algorithm. The error estimates in $L^2$ norm and energ…
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-orde…
With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jum…
The implementation of the finite element method for linear elliptic equations requires to assemble the stiffness matrix and the load vector. In general, the entries of this matrix-vector system are not known explicitly but need to be approximated by quadrature rules. If the coefficient functions of the differential operator or the forcing term are irregular, then standard quadrature formulas, such as the barycentric quadrature rule, may not be reliable. In this paper we investigate the application of two randomized quadrature formulas to the finite element method for such elliptic boundary val…
