Elliptic Differential Equations Research Articles

A simple proof of coerciveness of first-order system least-squares methods for general second-order elliptic PDEs

A simple proof of coerciveness of first-order system least-squares methods for general second-order elliptic PDEs

Emergence of biological transportation networks as a self-regulated process

We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal $L^2$-gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs.

Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls

Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls

Sparse optimal control of a quasilinear elliptic PDE in measure spaces

<p style='text-indent:20px;'>We prove existence of optimal controls for sparse optimal control of a quasilinear elliptic equation in measure spaces and derive first-order necessary optimality conditions. Under additional assumptions also second-order necessary and sufficient optimality conditions are obtained.</p>

Collocation approximation by deep neural ReLU networks for parametric and stochastic PDEs with lognormal inputs

We find the convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ in the noncompact set ${\mathbb R}^\infty$. The approximation error is measured in the norm of the Bochner space $L_2({\mathbb R}^\infty, V, \gamma)$, where $\gamma$ is the infinite tensor-product standard Gaussian probability measure on ${\mathbb R}^\infty$ and $V$ is the energy space. We also obtain similar dimension-independent results in the case when the lognormal inputs are parametrized by ${\mathbb R}^M$ of very large dimension $M$, and the approximation error is measured in the $\sqrt{g_M}$-weighted uniform norm of the Bochner space $L_\infty^{\sqrt{g}}({\mathbb R}^M, V)$, where $g_M$ is the density function of the standard Gaussian probability measure on ${\mathbb R}^M$. Bibliography: 62 titles.

Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs

Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs

A deep First-Order System Least Squares method for solving elliptic PDEs

A deep First-Order System Least Squares method for solving elliptic PDEs

MooAFEM: An object oriented Matlab code for higher-order adaptive FEM for (nonlinear) elliptic PDEs

We present an easily accessible, object oriented code (written exclusively in Matlab) for adaptive finite element simulations in 2D. It features various refinement routines for triangular meshes as well as fully vectorized FEM ansatz spaces of arbitrary polynomial order and allows for problems with very general coefficients. In particular, our code can handle problems typically arising from iterative linearization methods used to solve nonlinear PDEs. Due to the object oriented programming paradigm, the code can be used easily and is readily extensible. We explain the basic principles of our code and give numerical experiments that underline its flexibility as well as its efficiency.

Numerical Analysis of Fourier Finite Volume Element Method for Dirichlet Boundary Optimal Control Problems Governed by Elliptic PDEs on Complex Connected Domains

In this research, we investigate an optimal control problem governed by elliptic PDEs with Dirichlet boundary conditions on complex connected domains, which can be utilized to model the cooling process of concrete dam pouring. A new convergence result for two-dimensional Dirichlet boundary control is proven with the Fourier finite volume element method. The Lagrange multiplier approach is employed to find the optimality systems of the Dirichlet boundary optimal control problem. The discrete optimal control problem is then obtained by applying the Fourier finite volume element method based on Galerkin variational formulation for optimality systems, that is, using Fourier expansion in the azimuthal direction and the finite volume element method in the radial direction, respectively. In this way, the original two-dimensional problem is reduced to a sequence of one-dimensional problems, with the Dirichlet boundary acting as an interval endpoint at which a quadratic interpolation scheme can be implemented. The convergence order of state, adjoint state, and Dirichlet boundary control are therefore proved. The effectiveness of the method is demonstrated numerically, and numerical data is provided to support the theoretical analysis.

An integral equation method for the advection-diffusion equation on time-dependent domains in the plane

Boundary integral methods are attractive for solving homogeneous elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.

Entire downward solitons to the scalar curvature flow in Minkowski space

We investigate existence and uniqueness in Minkowski space of entire downward translating solitons with prescribed values at infinity, for a scalar curvature flow equation. The radial case translates into an ordinary differential equation and the general case into a fully non-linear elliptic PDE on \mathbb{R}^n .

Determining approximate solution of partial differential equations using radial basis function model trained with parallel symbiotic organisms search algorithm

SummaryPartial differential equations (PDEs) find extensive applications in geophysics (weather and climate modeling), astrophysics, and quantum mechanics. Solving PDEs by the conventional methods at times becomes difficult due to the complexity involved in these real‐world problems. Thus instead of an exact solution determining an approximate solution is also helpful for solving the problems. Here a model based on radial basis function (RBF) trained with parallel symbiotic organism search (PSOS) optimization is proposed to find out an approximate solution. The PSOS is a nature‐inspired optimization algorithm that ensures better accuracy due to the potential exploration of agents in the search space. Simultaneously, the computational complexity of this algorithm is low due to inbuilt parallelism. Simulation studies are reported for five real applications of PDEs: 2D modified Helmholtz equation, Poisson problem with Dirichlet boundary condition, elliptic PDE equation, and two convection‐diffusion equations. Comparison is carried out with the same RBF model parameters trained with parallel social spider optimization, original symbiotic organisms search algorithm, real coded genetic algorithm, and particle swarm optimization. The proposed model achieves minimum root mean square errors in most of the cases and achieves it with lower run time compared to the existing algorithms.

A fully-implicit hybridized discontinuous Galerkin spectral element method for two phase flow in petroleum reservoirs

A hybridized discontinuous Galerkin method with high order spectral element discretization is presented for modeling multiphase immiscible 2D fluid flow in petroleum reservoir simulations. The time dependent nonlinear governing equations for reservoir simulation are presented, and a fully coupled implicit time stepping procedure is implemented using a high order W-Rosenbrock algorithm. For incompressible fluids considered in the present study, the governing PDE's include an elliptic PDE for the pressure field and a parabolic convective diffusion equation for the saturation (water volume fraction) in a two phase oil-water system. The PDE systems are coupled through the highly nonlinear relative permeability and capillary pressure functions relating the fluid velocities of individual species to the individual species pressures and saturations. With HDG/spectral element (HDGSEM) spatial discretization of the coupled governing equations, the system constitutes an index 1 differential algebraic equation. It is shown that the W-Rosenbrock algorithm in combination with the HDGSEM method yields an efficient and effective solution procedure which demonstrates robust, high order temporal accuracy.Numerous physical tests cases are presented demonstrating that the versatility of this approach is sufficient to address the challenges in modeling heterogeneous permeability fields and geologic features of arbitrary shape with discontinuous permeability functions. It is also capable of resolving the sharp time dependent displacement fronts in oil-water systems encountered in secondary oil recovery. It is demonstrated that the high order polynomial basis functions (up to 12th order) used in the HDGSEM algorithm have the ability to resolve complex geometry and achieve accurate results utilizing a simple Cartesian mesh for non-conforming geological features. Eliminating the need to re-grid and conform the mesh to the boundaries of different geological features greatly simplifies the workflow for petroleum engineers.

A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components’ interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component – associated with the solution on interior boundaries – and a bubble component that vanishes at ports: since the bubble components are uniquely determined by the solution value at the corresponding port, we can recast the constrained optimization statement into an unconstrained statement, which reads as a nonlinear least-squares problem and can be solved using the Gauss–Newton method. We present thorough numerical investigations for a two-dimensional neo-Hookean nonlinear mechanics problem to validate our method; we further discuss the well-posedness of the mathematical formulation and the a priori error analysis for linear coercive problems.

TTRISK: Tensor train decomposition algorithm for risk averse optimization

AbstractThis article develops a new algorithm named TTRISK to solve high‐dimensional risk‐averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so‐called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large‐scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID‐19. The results indicate that the risk‐averse framework is feasible with the tensor approximations under tens of random variables.

On the boundary regularity of solutions of linear elliptic partial differential equations

Regularity refers to the properties of the solution, including the smoothness, symmetry, and asymptotic of the solution. It is an important part of the theoretical study of partial differential equations. It plays a key role in the existence, uniqueness, stability, and smoothness of the theoretical solutions of partial differential equations. It is an important basis for understanding the nature of partial differential equations and their corresponding physical reality. This paper studies the boundary regularity of elliptic partial differential equations, including the problem of the oblique boundary of completely nonlinear equations. It is well known that the regularity of the solution at the region boundary depends not only on the equation, but also on the geometric properties of the region boundary. This is why boundary regularity is complicated. This paper is to obtain the regularity of the solution at the region boundary under different boundary conditions.

A localized Fourier collocation method for 2D and 3D elliptic partial differential equations: Theory and MATLAB code

AbstractA localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems. The method first discretizes the entire domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated using the pseudo‐spectral Fourier collocation method. The key idea of the present method is to combine the merits of the quick convergence of the pseudo‐spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large‐scale simulations. The present method can be viewed as a competitive alternative for solving numerically large‐scale boundary value problems with complex‐shape geometries. Preliminary numerical experiments involving Poisson, Helmholtz, and modified‐Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.

Free boundary regularity for a class of one-phase problems with non-homogeneous degeneracy

We consider a one-phase free boundary problem governed by doubly degenerate fully nonlinear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the theory of non-autonomous integrals. By way of brief elucidating example, such nonlinear problems in force appear in the mathematical theory of combustion, as well as in the study of some flame propagation problems. In such an environment we prove that solutions are Lipschitz continuous and they fulfil a non-degeneracy property. Furthermore, we address the Caffarelli’s classification scheme: Flat and Lipschitz free boundaries are locally C1,β for some 0 < β(universal) < 1. Particularly, our findings are new even in the toy model $${{\cal G}_{p,q}}\left[u \right]: = \left[{{{\left| {\nabla u} \right|}^p} + \mathfrak{a}\left(x \right){{\left| {\nabla u} \right|}^q}} \right]\Delta u,\,\,\,\,\,{\rm{for}}\,\,0 < p < q {{\cal G}_{p,q}}\left[u \right]: = \left[{{{\left| {\nabla u} \right|}^p} + \mathfrak{a}\left(x \right){{\left| {\nabla u} \right|}^q}} \right]\Delta u,\,\,\,\,\,{\rm{for}}\,\,0 < p < q < \infty \,\,{\rm{and}}\,\,0 \le \mathfrak{a} \in {C^0}\left(\Omega \right) \infty \,\,{\rm{and}}\,\,0 \le \mathfrak{a} \in {C^0}\left(\Omega \right)$$ We also bring to light other interesting doubly degenerate settings where our results still work. Finally, we present some key tools in the theory of degenerate fully nonlinear PDEs, which may have their own mathematical importance and applicability.

Partial Differential Equations

The workshop covered topics in nonlinear elliptic and parabolic Partial Differential Equations as well as topics in Geometric Measure Theory, touching topics such as geometric variational problems and minimal surfaces, geometric flows, free boundaries and the structure of nodal sets of eigenfunctions as well as real and complex Monge–Ampère equations.

Two-Scale Asymptotic Homogenization Method for Composite Kirchhoff Plates with in-Plane Periodicity

This paper develops a two-scale asymptotic homogenization method for periodic composite Kirchhoff plates. In this method, a three-dimensional (3D) periodic plate problem is simplified as a Kirchhoff plate problem, which is governed by a fourth-order uniformly elliptic partial differential equation (PDE) with periodically oscillating coefficients. Then, a two-scale solution in an asymptotic expansion form is presented for the PDE, and it is found that the first-order perturbed displacement in the asymptotic solution is zero. Additionally, periodic boundary and normalization constraint conditions are proposed to determine the unique solution to unit cell problems. Moreover, standard finite element formulations for calculating the perturbed displacements are derived from the principle of virtual work. Physical interpretations of the influence functions are presented by analyzing the properties of self-balanced quasi-load vectors used for solving the influence functions. Numerical comparisons show that the present method is physically acceptable and highly accurate.