Elliptic Differential Equations Research Articles

On the use of elliptic PDEs for the parameterisation of planar multipatch domains

AbstractThis paper presents a parameterisation framework based on (inverted) elliptic PDEs for addressing the planar parameterisation problem of finding a valid description of the domain’s interior given no more than a spline-based description of its boundary contours. The framework is geared towards isogeometric analysis (IGA) applications wherein the physical domain is comprised of more than four sides, hence requiring more than one patch. We adopt the concept of harmonic maps and propose several PDE-based problem formulations capable of finding a valid map between a convex parametric multipatch domain and the piecewise-smooth physical domain with an equal number of sides. In line with the isoparametric paradigm of IGA, we treat the parameterisation problem using techniques that are characteristic for the analysis step. As such, this study proposes several IGA-based numerical algorithms for the problem’s governing equations that can be effortlessly integrated into a well-developed IGA software suite. We augment the framework with mechanisms that enable controlling the parametric properties of the outcome. Parametric control is accomplished by, among other techniques, the introduction of a curvilinear coordinate system in the convex parametric domain, for which more general elliptic PDEs are adopted. Depending on the application, parametric control allows for building desired features into the computed map, such as homogeneous cell sizes or boundary layers.

Stability of the isotropic conductivity reconstruction using magnetic resonance electrical impedance tomography (MREIT)

Abstract Magnetic resonance electrical impedance tomography (MREIT) is a high-resolution imaging modality that aims to reconstruct the objects' conductivity distributions at low frequency using the measurable $z$-th component of the magnetic flux density obtained from an MRI scanner. Traditional reconstruction algorithms in MREIT use two data subject to two linearly independent current densities. However, the temporal resolution of such a MREIT image is relatively low. Recently, a single current MREIT has been proposed to improve the temporal resolution. Even though a series of reconstruction algorithms have been proposed in the last two decades, the theoretical studies of MREIT are still quite limited. This paper presents the stability theorems for two datum and a single data-based isotropic conductivity reconstruction using MREIT. Using the regularity theory of elliptic partial differential equations, we prove that the only instability in the inverse problem of MREIT comes from taking the second-order derivative of the measured data $B_z$, the $z$-th component of the magnetic flux density. To get a stable reconstruction from the noisy $B_z$ data, we note that the edge structure of $\nabla B_z$ reveals the edge features in the unknown conductivity and provides an edge-preserving denoising approach for the $\nabla B_z$ data. We use a modified Shepp-Logan phantom model to validate the proposed theory and the denoising approach.

On the Coincidence between Campanato Functions and Lipschitz Functions: A New Approach via Elliptic PDES

Abstract Let $({\mathcal{M}},d,\mu)$ be the metric measure space with a Dirichlet form $\mathscr{E}$. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on ${\mathcal{M}}$ to its Poisson integral Ptu on ${\mathcal{M}}\times\mathbb{R}_+$. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math., 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971].

Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

Simulations and analysis of underwater acoustic wave propagation model based on Helmholtz equation

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L 2 and L ∞ norms.

Solving a class of multi-scale elliptic PDEs by Fourier-based mixed physics informed neural networks

Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential equations (PDEs) with multiple scales by utilizing Fourier-based mixed physics informed neural networks (dubbed FMPINN), its solver is configured as a multi-scale deep neural network. In contrast to the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix caused by the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, to validate the efficiency and robustness of the proposed FMPINN algorithm, we present several numerical examples of multi-scale problems in various dimensional Euclidean spaces. These examples cover low-frequency and high-frequency oscillation cases, demonstrating the effectiveness of our approach. All code and data accompanying this manuscript will be publicly available at https://github.com/Blue-Giant/FMPINN.

A Numerical Expedition through the Mathematical Representation of Complex Braided Morphometry—A Case Study of Brahmaputra River in India

The present work explores the process of mathematical representation for the complex geometry of a wide alluvial river with high braiding intensities. It primarily focuses on an approach to developing a numerical solution algorithm for representing the complex channel geometry of the braided Brahmaputra River. Traditional elliptic PDEs with boundary-fitted coordinate transformation were deployed, converting the non-uniform physical plane into a transformed uniform orthogonal computational plane. This study was conducted for the river channel reach with upstream and downstream nodes at Pandu and Jogighopa (reach length ~100 km), respectively, within the Assam flood plain in India, with fourteen measured river cross-sections for the year of 1997. The geo-referenced image covering the river stretch in 1997 was delineated using a ArcGIS software 9.0 tool by digitizing the bank lines. Stream bed interpolation was conducted by interpolating bed elevation from a bathymetrical database onto code-generated mesh nodes. Discretization of the domain was performed through the developed computer code, and the bed-level matrix was generated by the IDW method as well as the MATLAB tool using the nearest neighborhood technique. A mathematical representation of a digital terrain model was thus developed. This generated model was employed as a geometrical data input to simulate secondary flow utilizing 2D depth-averaged equations with the flow dispersion stress tensor as an extra source component, coming from curvilinear flow patterns caused by severe river braiding. The developed model may further be useful in mathematically representing the geometrical complexities of braided rivers with a relatively realistic assessment of the various parameters involved if deployed with improved river modeling with morphometric evolution.

An optimization-based method for sign-changing elliptic PDEs

We study the numerical approximation of sign-shifting problems of elliptic type. We fully analyze and assess the method briefly introduced in [Abdulle, Huber, Lemaire; CRAS, 17]. Our method is based on domain decomposition and optimization. Upon an extra integrability assumption on the exact normal flux trace along the sign-changing interface, our method is proved to be convergent as soon as, for a given loading, the PDE admits a unique solution of finite energy. Departing from the T-coercivity approach, which relies on the use of geometrically fitted mesh families, our method works for arbitrary (interface-compliant) mesh sequences. Moreover, it is shown convergent for a class of problems for which T-coercivity is not applicable. A comprehensive set of test-cases complements our analysis.

Multinode Shepard method for two-dimensional elliptic boundary problems on different shaped domains

In this paper, we continue the study on the application of multinode Shepard method to numerically solve elliptic Partial Differential Equations (PDEs) equipped with various conditions at the boundary of domains of different shapes. In particular, for the first time, the multinode Shepard method is proposed to solve elliptic PDEs with Dirichlet and/or Neumann boundary conditions. The method has been opportunely handled to efficiently work dealing with scattered distribution of points and, to this aim, several experiments in different 2d domains have been performed. Comparisons with the analytic solution and the results generated by the Kansa’s RBF solvers have been reported referring to Halton points. The results are very promising and should be of interest for applications in the real world.

Sparse Spectral Methods for Solving High-Dimensional and Multiscale Elliptic PDEs

Sparse Spectral Methods for Solving High-Dimensional and Multiscale Elliptic PDEs

An Explicit Form of Harnack’s Inequality for Poisson’s Dirichlet Problem

The explicit form of Harnack’s inequality for non-negative harmonic functions in the open ball plays an important role in harmonic analysis and elliptic partial differential equations. In this paper, we establish the explicit form of Harnack’s inequality for Poisson’s Dirichlet problem in the open ball with non-negative boundary data. We involve the Green’s function for Laplacian operator to find it out. Harnack’s inequality for positive harmonic functions can be followed from Harnack’s inequality for Poisson’s Dirichlet problem when the source function is set to be zero.

Solution of Laplace Equation by Modified Differential Transform Method

In this paper, we applied the modified two-dimensional differential transform method to solve Laplace equation. Laplace equation is one of Elliptic partial differential equations. These kinds of differential equations have specific applications models of physics and engineering. We consider four models with two Dirichlet and two Neumann boundary conditions. The simplicity of this method compared to other iteration methods is shown here. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.

Robust domain decomposition methods for high-contrast multiscale problems on irregular domains with virtual element discretizations

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains.The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under [Formula: see text]-Gevrey assumptions on the residual equation, we establish [Formula: see text]-Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

CUQIpy: II. Computational uncertainty quantification for PDE-based inverse problems in Python

Inverse problems, particularly those governed by Partial Differential Equations (PDEs), are prevalent in various scientific and engineering applications, and uncertainty quantification (UQ) of solutions to these problems is essential for informed decision-making. This second part of a two-paper series builds upon the foundation set by the first part, which introduced CUQIpy, a Python software package for computational UQ in inverse problems using a Bayesian framework. In this paper, we extend CUQIpy’s capabilities to solve PDE-based Bayesian inverse problems through a general framework that allows the integration of PDEs in CUQIpy, whether expressed natively or using third-party libraries such as FEniCS. CUQIpy offers concise syntax that closely matches mathematical expressions, streamlining the modeling process and enhancing the user experience. The versatility and applicability of CUQIpy to PDE-based Bayesian inverse problems are demonstrated on examples covering parabolic, elliptic and hyperbolic PDEs. This includes problems involving the heat and Poisson equations and application case studies in electrical impedance tomography and photo-acoustic tomography, showcasing the software’s efficiency, consistency, and intuitive interface. This comprehensive approach to UQ in PDE-based inverse problems provides accessibility for non-experts and advanced features for experts.

Liouville theorems and optimal regularity in elliptic equations

AbstractThe objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic partial differential equations with measurable coefficients and the Liouville property at infinity. Initially, we address the two‐dimensional case by proving an Alt–Caffarelli–Friedman‐type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole‐filling technique, we present a distinct “almost‐monotonicity” formula that implies Hölder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow‐up and ‐convergence arguments.

Stability and critical dimension for Kirchhoff systems in closed manifolds

Abstract The Kirchhoff equation was proposed in 1883 by Kirchhoff [Vorlesungen über Mechanik, Leipzig, Teubner, 1883] as an extension of the classical D’Alembert’s wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions [“On some questions in boundary value problems of mathematical physics,” in Contemporary Developments in Continuum Mechanics and PDE’s, G. M. de la Penha, and L. A. Medeiros, Eds., Amsterdam, North-Holland, 1978] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as ∂ 2 u ∂ t 2 + a + b ∫ Ω | ∇ u | 2 d x Δ u = f ( x , u ) , $\frac{{\partial }^{2}u}{\partial {t}^{2}}+\left(a+b{\int }_{{\Omega}}\vert \nabla u{\vert }^{2}\mathrm{d}x\right){\Delta}u=f\left(x,u\right),$ where Δ = − ∑ ∂ 2 ∂ x i 2 ${\Delta}=-\sum \frac{{\partial }^{2}}{\partial {x}_{i}^{2}}$ is the Laplace-Beltrami Euclidean Laplacian. We investigate in this paper a closely related stationary version of this equation, in the case of closed manifolds, when u is vector valued and when f is a pure critical power nonlinearity. We look for the stability of the equations we consider, a question which, in modern nonlinear elliptic PDE theory, has its roots in the seminal work of Gidas and Spruck.

A posterior contraction for Bayesian inverse problems in Banach spaces

This paper features a study of statistical inference for linear inverse problems with Gaussian noise and priors in structured Banach spaces. Employing the tools of sectorial operators and Gaussian measures on Banach spaces, we overcome the theoretical difficulty of lacking the bias-variance decomposition in Banach spaces, characterize the posterior distribution of solution though its Radon–Nikodym derivative, and derive the optimal convergence rates of the corresponding square posterior contraction and the mean integrated square error. Our theoretical findings are applied to two scenarios, specifically a Volterra integral equation and an inverse source problem governed by an elliptic partial differential equation. Our investigation demonstrates the superiority of our approach over classical results. Notably, our method achieves same order of convergence rates for solutions with reduced smoothness even in a Hilbert setting.

A fully implicit edge/face centered discontinuous Galerkin / mixed finite element scheme for the advection-dispersion equation

Mixed Finite Element (MFE) method is a robust numerical technique for solving elliptic and parabolic partial differential equations (PDEs). However, MFE can generate solutions with strong unphysical oscillations and/or large numerical diffusion for hyperbolic type PDEs. For its part, Discontinuous Galerkin (DG) finite element method is well adapted to solve hyperbolic systems and can accurately reproduce solutions involving sharp fronts. Therefore, the combination of DG and MFE is a good strategy for solving hyperbolic/parabolic problems such as advection – diffusion/dispersion equations. The classical formulation of the two methods is based on operator and time splitting allowing for separate solutions to advection with an explicit scheme and to dispersion with an implicit scheme. However, this kind of approach has the following drawbacks: (i) it lacks efficiency, as two systems with different unknowns are solved at each time step, (ii) it induces errors generated by the splitting, (iii) it can be CPU wise-expensive because of the CFL constraint, and (iv) it cannot be employed for steady-state transport simulations.To overcome these difficulties, we develop in this work a fully implicit edge/face centered DG-MFE formulation where the two methods share the same unknowns. In this formulation, the DG method is developed on lumping regions associated with the mesh edges/faces instead of mesh elements. Thus, the traces of concentration at mesh edges/faces, which are the Degrees Of Freedom (DOF) of the hybrid-MFE, are also part of the DOFs of the DG. The temporal discretization is based on the Crank–Nicolson method for both advection and dispersion. Numerical tests are performed to validate the new scheme by comparison against an analytical solution and to show its ability to handle steady-state transport simulations.The procedure is developed for 2D triangular meshes but can easily be extended to other 2D and 3D shape elements.

On the Solvability of an Essentially Nonlinear Elliptic Differential Equation with Nonlocal Boundary Conditions

On the Solvability of an Essentially Nonlinear Elliptic Differential Equation with Nonlocal Boundary Conditions