In this paper, we present proofs of the coerciveness of first-order system least-squares methods for general (possibly indefinite) second-order linear elliptic PDEs under a minimal uniqueness assumption. For general linear second-order elliptic PDEs, the uniqueness, existence, and well-posedness are equivalent due to the compactness of the operator and Fredholm alternative. Thus only a minimal uniqueness assumption is assumed: the homogeneous equation has a unique zero solution. The coerciveness of the standard variational problem is not required. The paper's main contribution is our first proof, which is a straightforward and short proof using the inf-sup stability of the standard variational formulation. The proof can potentially be applied to other equations or settings once having the standard formulation's stability. We also present two other proofs for the least-squares methods of general second-order linear elliptic PDEs. The second proof is based on a lemma introduced in the discontinuous Petrov-Galerkin method, and the third proof is based on various stability analyses of the decomposed problems. As an application, we also discuss least-squares finite element methods for problems with a nonsingular $H^{-1}$ right-hand side.
In this paper, we propose a Galerkin finite element method for the elliptic optimal control problem governed by the Riesz space-fractional PDEs on 2D domains with control variable being discretized by variational discretization technique. The optimality condition is derived and priori error estimates of control, costate and state variables are successfully established. Numerical test is carried out to illustrate the accuracy performance of this approach.
We formulate and analyze a goal-oriented adaptive finite element method (GOAFEM) for a semilinear elliptic PDE and a linear goal functional. The strategy involves the finite element solution of a linearized dual problem, where the linearization is part of the adaptive strategy. Linear convergence and optimal algebraic convergence rates are shown.
In this work, we derive a priori error estimate of the mixed residual method when solving some elliptic PDEs. Our work is the first theoretical study of this method. We prove that the neural network solutions will converge if we increase the training samples and network size without any constraint on the ratio of training samples to the network size. Besides, our results suggest that the mixed residual method can recover high order derivatives better than the deep Ritz method, which has also been verified by our numerical experiments.
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…
Based on the Fourier extension, we propose an oversampling collocation method for solving the elliptic partial differential equations with variable coefficients over arbitrary irregular domains. This method only uses the function values on the equispaced nodes, which has low computational cost and versatility. While a variety of numerical experiments are presented to demonstrate the effectiveness of this method, it shows that the approximation error fast reaches a plateau with increasing the degrees of freedom, due to the inherent ill-conditioned of frames.
This paper establishes the optimal $H^1$-norm error estimate for a nonstandard finite element method for approximating $H^2$ strong solutions of second order linear elliptic PDEs in non-divergence form with continuous coefficients. To circumvent the difficulty of lacking an effective duality argument for this class of PDEs, a new analysis technique is introduced; the crux of it is to establish an $H^1$-norm stability estimate for the finite element approximation operator which mimics a similar estimate for the underlying PDE operator recently established by the authors and its proof is based o…
In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain assumptions on the nonlinear coefficient, we show that the discrete problem has a unique solution. This is achieved by showing that the associated operator satisfies certain continuity and monotonicity properties. With the help of these properties, we derive optimal error estimates in the energy norm. We present several numerical examples to verify the error…
We develop a boundary integral equation solver for elliptic partial differential equations on complex \threed geometries. Our method is efficient, high-order accurate and robustly handles complex geometries. A key component is our singular and near-singular layer potential evaluation scheme, \qbkix: a simple extrapolation of the solution along a line to the boundary. We present a series of geometry-processing algorithms required for \qbkix to run efficiently with accuracy guarantees on arbitrary geometries and an adaptive upsampling scheme based on a iteration-free heuristic for quadrature err…
We obtained convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ from the non-compact set $\mathbb{R}^\infty$. The approximation error is measured in the norm of the Bochner space $L_2(\mathbb{R}^\infty, V, γ)$, where $γ$ is the infinite tensor product standard Gaussian probability measure on $\mathbb{R}^\infty$ and $V$ is the energy space. We also obtained similar results for the case when the lognormal inputs are parametrized on $\mathbb{R}^M$ with very large dimension $M$, and th…
In this paper we introduce an algorithm based on a sparse grid adaptive refinement, for the approximation of the eigensolutions to parametric problems arising from elliptic partial differential equations. In particular, we are interested in detecting the crossing of the hypersurfaces describing the eigenvalues as a function of the parameters. The a priori matching is followed by an a posteriori verification, driven by a suitably defined error indicator. At a given refinement level, a sparse grid approach is adopted for the construction of the grid of the next level, by using the marking given …
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 …
An EDG Method for Distributed Optimal Control of Elliptic PDEs
College of Mathematics, Sichuan University（四川大学数学学院）；Department of Mathematics and Statistics, Missouri University of Science and Technology（密苏里科技大学数学与统计系）
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…
Sparse polynomial approximation for optimal control problems constrained by elliptic PDEs with lognormal random coefficients
Oden Institute for Computational Engineering & Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；Department of Mechanical Engineering, and Department of Geological Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校机械工程系和地质科学系）
In this paper, we propose a certified reduced basis (RB) method for quasilinear elliptic problems together with its application to nonlinear magnetostatics equations, where the later model permanent magnet synchronous motors (PMSM). The parametrization enters through the geometry of the domain and thus, combined with the nonlinearity, drives our reduction problem. We provide a residual-based a-posteriori error bound which, together with the Greedy approach, allows to construct reduced-basis spaces of small dimensions. We use the empirical interpolation method (EIM) to guarantee the efficient o…
This paper studies numerical methods for the approximation of elliptic PDEs with lognormal coefficients of the form $-{\rm div}(a\nabla u)=f$ where $a=\exp(b)$ and $b$ is a Gaussian random field. The approximant of the solution $u$ is an $n$-term polynomial expansion in the scalar Gaussian random variables that parametrize $b$. We present a general convergence analysis of weighted least-squares approximants for smooth and arbitrarily rough random field, using a suitable random design, for which we prove optimality in the following sense: their convergence rate matches exactly or closely the ra…
General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. …
In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with $\mathrm{…
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indica…
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…
