Second-order PDEs Research Articles

Imaging Anisotropic Conductivities from Current Densities

In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(\Omega)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(\Omega)^{d,d}$-norms, the existence of minimizers and optimality systems of the regularized formulation using the concept of H-convergence. Further, we provide a detailed analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of H-convergence. In addition, we develop a projected Newton algorithm for solving the first-order optimality system. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.

MINRES for Second-Order PDEs with Singular Data

Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov--Galerkin method with optimal test functions (DPG) usually exclude singular data, e.g., non square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of $H^{-1}$ loads, and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.

“Mixed” Meshless Time-Domain Adaptive Algorithm for Solving Elasto-Dynamics Equations

A time-domain adaptive algorithm was developed for solving elasto-dynamics problems through a mixed meshless local Petrov-Galerkin finite volume method (MLPG5). In this time-adaptive algorithm, each time-dependent variable is interpolated by a time series function of n-order, which is determined by a criterion in each step. The high-order series of expanded variables bring high accuracy in the time domain, especially for the elasto-dynamic equations, which are second-order PDE in the time domain. In the present mixed MLPG5 dynamic formulation, the strains are interpolated independently, as are displacements in the local weak form, which eliminates the expensive differential of the shape function. In the traditional MLPG5, both shape function and its derivative for each node need to be calculated. By taking the Heaviside function as the test function, the local domain integration of stiffness matrix is avoided. Several numerical examples, including the comparison of our method, the MLPG5–Newmark method and FEM (ANSYS) are given to demonstrate the advantages of the presented method: (1) a large time step can be used in solving a elasto-dynamics problem; (2) computational efficiency and accuracy are improved in both space and time; (3) smaller support sizes can be used in the mixed MLPG5.

Conservation of Forces and Total Work at the Interface Using the Internodes Method

The Internodes method is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into disjoint subdomains. In this paper we are interested in measuring how much the Internodes method is conservative across the interface. If hp-fem discretizations are employed, we prove that both the total force and total work generated by the numerical solution at the interface of the decomposition vanish in an optimal way when the mesh size tends to zero, i.e., like mathcal {O}(h^{p}), where p is the local polynomial degree and h the mesh-size. This is the same as the error decay in the H1-broken norm. We observe that the conservation properties of a method are intrinsic to the method itself because they depend on the way the interface conditions are enforced rather then on the problem we are called to approximate. For this reason, in this paper, we focus on second-order elliptic PDEs, although we use the terminology (of forces and works) proper of linear elasticity. Two and three dimensional numerical experiments corroborate the theoretical findings, also by comparing Internodes with Mortar and WACA methods.

Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEs

For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients {mathbf {A}}, {mathbf {b}},gamma in L^infty and symmetric and uniformly positive definite coefficient matrix {mathbf {A}}, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in H({{,mathrm{div},}})times L^2 as well as in in L^2times L^2 up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to L^infty coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not apply immediately to the mixed formulation in H({{,mathrm{div},}})times L^2. But it allows the uniform approximation of some L^2 contributions and can be combined with a recent L^2 best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.

Solving some stochastic partial differential equations driven by Lévy noise using two SDEs*

ABSTRACT The method of characteristics is a powerful tool to solve some nonlinear second-order stochastic PDEs like those satisfied by a consistent dynamic utilities, see [N. El Karoui and M. Mrad, An exact connection between two solvable SDEs and a non linear utility stochastic PDEs, SIAM J. Financ. Math. 4(1) (2013), pp. 737–783; A. Matoussi and M. Mrad, Dynamic utility and related nonlinear SPDE driven by Lévy noise, preprint (2020), submitted for publication. Available at https://hal.archives-ouvertes.fr/hal-03025475]. In this situation the solution is theoretically of the form where and are solutions of a system of two SDEs, is the inverse flow of and is the initial condition. Unfortunately this representation is not explicit except in simple cases where and are solutions of linear equations. The objective of this work is to take advantage of this representation to establish a numerical scheme approximating the solution V using Euler approximations and of X and ξ. This allows us to avoid a complicated discretization in time and space of the SPDE for which it seems really difficult to obtain error estimates. We place ourselves in the framework of SDEs driven by Lévy noise and we establish at first a strong convergence result, in -norms, of the compound approximation to the compound variable , in terms of the approximations of X and Y which are solutions of two SDEs with jumps. We then apply this result to Utility-SPDEs of HJB type after inverting monotonic stochastic flows.

Stochastic optimal control with random coefficients and associated stochastic Hamilton\u2013Jacobi\u2013Bellman equations

We consider the optimal control problem for stochastic differential equations (SDEs) with random coefficients under the recursive-type objective functional captured by the backward SDE (BSDE). Due to the random coefficients, the associated Hamilton–Jacobi–Bellman (HJB) equation is a class of second-order stochastic PDEs (SPDEs) driven by Brownian motion, which we call the stochastic HJB (SHJB) equation. In addition, as we adopt the recursive-type objective functional, the drift term of the SHJB equation depends on the second component of its solution. These two generalizations cause several technical intricacies, which do not appear in the existing literature. We prove the dynamic programming principle (DPP) for the value function, for which unlike the existing literature we have to use the backward semigroup associated with the recursive-type objective functional. By the DPP, we are able to show the continuity of the value function. Using the Itô–Kunita’s formula, we prove the verification theorem, which constitutes a sufficient condition for optimality and characterizes the value function, provided that the smooth (classical) solution of the SHJB equation exists. In general, the smooth solution of the SHJB equation may not exist. Hence, we study the existence and uniqueness of the solution to the SHJB equation under two different weak solution concepts. First, we show, under appropriate assumptions, the existence and uniqueness of the weak solution via the Sobolev space technique, which requires converting the SHJB equation to a class of backward stochastic evolution equations. The second result is obtained under the notion of viscosity solutions, which is an extension of the classical one to the case for SPDEs. Using the DPP and the estimates of BSDEs, we prove that the value function is the viscosity solution to the SHJB equation. For applications, we consider the linear-quadratic problem, the utility maximization problem, and the European option pricing problem. Specifically, different from the existing literature, each problem is formulated by the generalized recursive-type objective functional and is subject to random coefficients. By applying the theoretical results of this paper, we obtain the explicit optimal solution for each problem in terms of the solution of the corresponding SHJB equation.

Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

Einstein–Weyl geometry is a triple ({mathbb {D}},g,omega ) where {mathbb {D}} is a symmetric connection, [g] is a conformal structure and omega is a covector such that bullet connection {mathbb {D}} preserves the conformal class [g], that is, {mathbb {D}}g=omega g; bullet trace-free part of the symmetrised Ricci tensor of {mathbb {D}} vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector omega is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector omega is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and omega provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE

Abstract We investigate the problem with local homogeneous two-point conditions with respect to time for nonhomogeneous PDE of second order in time variable and generally infinite order in spatial variables in the case when the characteristic determinant of the problem identically equals zero. We establish the nonexistence conditions of solution of this problem in the class of entire functions.

PDE-based hyperbolic-parabolic model for image denoising with forward-backward diffusivity

In the present study, we propose an effective nonlinear anisotropic diffusion-based hyperbolic parabolic model for image denoising and edge detection. The hyperbolicparabolic model employs a second-order PDEs and has a second-time derivative to time t. This approach is very effective to preserve sharper edges and better-denoised images of noisy images. Our model is well-posed and it has a unique weak solution under certain conditions, which is obtained by using an iterative finite difference explicit scheme. The results are obtained in terms of peak signal to noise ratio (PSNR) as a metric, using an explicit scheme with forward-backward diffusivities.

Hamiltonian elliptic systems with critical polynomial-exponential growth

In this work, we revisit a classical problem of Hamiltonian-type systems of second-order elliptic PDE in bounded domains. Recalling the famous critical hyperbola associated with this problem, we propose an approach where one of the nonlinearities has a polynomial growth over this hyperbola’s asymptotes, which allows the other one to have exponential growth. We establish solutions for these critical systems by using the reduction by inversion method and some new results for sharp Adams Inequality and Concentration–Compactness for high order N-Laplace operators.

Second-Order PDE Constrained Controlled Optimization Problems with Application in Mechanics

The present paper deals with a class of second-order PDE constrained controlled optimization problems with application in Lagrange–Hamilton dynamics. Concretely, we formulate and prove necessary conditions of optimality for the considered class of control problems driven by multiple integral cost functionals involving second-order partial derivatives. Moreover, an illustrative example is provided to highlight the effectiveness of the results derived in the paper. In the final part of the paper, we present an algorithm to summarize the steps for solving a control problem such as the one investigated here.

Rate optimality of adaptive finite element methods with respect to overall computational costs

We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the adaptive mesh-refinement as well as the inexact solution of the arising discrete systems. We prove that the proposed strategy leads to linear convergence with optimal algebraic rates. Unlike prior works, however, we focus on convergence rates with respect to the overall computational costs. In explicit terms, the proposed adaptive strategy thus guarantees quasi-optimal computational time. In particular, our analysis covers linear problems, where the linear systems are solved by an optimally preconditioned CG method as well as nonlinear problems with strongly monotone nonlinearity which are linearized by the so-called Zarantonello iteration.

On certain degenerate and singular elliptic PDEs III: Nondivergence form operators and RH∞-weights

A Harnack inequality is proved for nonnegative strong solutions to certain second-order elliptic PDEs with rough coefficients whose degeneracies/singularities are shaped by weights in the reverse-Hölder class RH∞. Such operators include subelliptic Grushin and Grushin-like PDEs with unbounded first-order terms.

Adaptive Mixed Finite Element Methods for Non-self-adjoint Indefinite Second-Order Elliptic PDEs with Optimal Rates

This paper establishes the convergence of adaptive mixed finite element methods for second-order linear non-self-adjoint indefinite elliptic problems in three dimensions with piecewise Lipschitz co...

Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions

In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove that the proposed multiscale method achieves nearly exponential convergence in the approximation error with respect to the computational degrees of freedom. Our strategy is to perform an energy orthogonal decomposition of the solution space into a coarse scale component comprising $a$-harmonic functions in each element of the mesh, and a fine scale component named the bubble part that can be computed locally and efficiently. The coarse scale component depends entirely on function values on edges. Our approximation on each edge is made in the Lions--Magenes space $H_{00}^{1/2}(e)$, which we will demonstrate to be a natural and powerful choice. We construct edge basis functions using local oversampling and singular value decomposition. When local information of the right-hand side is adaptively incorporated into the edge basis functions, we prove a nearly exponential convergence rate of the approximation error. Numerical experiments validate and extend our theoretical analysis; in particular, we observe no obvious degradation in accuracy for high-contrast media problems.

Solutions of Second-Order PDEs with First-Order Quotients

We investigate a general approach for solving a partial differential equation by using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential invariants, we obtain new differential constraints which are compatible with the PDE under consideration. Adding these constraints to our system makes it overdetermined, and easier to solve. We focus on second-order scalar PDEs on a function of two variables whose quotients are first-order scalar PDEs. This situation occurs only when the Lie algebra of symmetries of the second-order PDE is infinite-dimensional. We apply this idea to various second-order PDEs with infinite-dimensional symmetry Lie algebras, one of which is the Hunter-Saxton equation.

Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. If the diffusion coefficient is semilinear (i.e, linear in the gradient of the solution and nonlinear in the solution; the drift can still be fully nonlinear), we establish a complete theory, including global existence and a comparison principle.

A Dufort–Frankel scheme for one-dimensional uncertain heat equation

Uncertain heat equation (for short, UHE) is a type of second-order uncertain PDEs driven by Liu processes. In most cases, since it is tough to get the analytic solution for a UHE, we must find a way to obtain its numerical solution. A forward difference scheme has been designed to solve UHE, but our paper will show that this method may exhibit instability in some situations. We explore another approach, unconditionally stable and namely Dufort–Frankel method. Moreover, this paper will use Dufort–Frankel method to calculate the expected value and extreme value of the solution for a UHE.

Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

In the present work, we consider weakly-singular integral equations arising from linear second-order elliptic PDE systems with constant coefficients, including, e.g. linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee that the weighted-residual error estimator is reliable and converges at optimal algebraic rate if used within an adaptive algorithm. These conditions are satisfied, e.g. for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include the localization of (non-local) fractional Sobolev norms and local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.