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Practical problems in scientific computation that solve differential equations rarely have explicit exact solutions. Therefore, verifying the correctness of such programs has long been a challenge due to the difficulty of producing expected outputs on test cases. In this paper, the principles of metamorphic testing are applied to verify programs that solve second‐order elliptic differential equations. We present a testing process specifically tailored for the verification testing of scientific computation programs and integrate it to the process of developing scientific software. Unlike existing approaches, we formally derive metamorphic relations from the numerical models of differential equations built in development process of scientific computing programs. The experimental results clearly show that our approach is effective in detecting faults commonly found in scientific computing programs. It outperforms the fault detecting ability of the trend method, which is a traditional testing method for scientific software. [ABSTRACT FROM AUTHOR]

We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO coefficients and a result for linear equations with Dini continuous coefficients. In addition, we provide an abstract theorem conditional on PDE estimates available. The linear results have the full range of weighted estimates with Muckenhoupt weights as a consequence. [ABSTRACT FROM AUTHOR]

We consider strongly elliptic differential-difference equations with mixed boundary conditions in a bounded domain. There are homogeneous Dirichlet conditions on a part of the boundary, and boundary conditions of the third kind on the other part of the boundary. We establish a connection between these problems and nonlocal mixed problems for strongly elliptic differential equations. We prove the uniqueness and the smoothness of their solutions. [ABSTRACT FROM AUTHOR]

The article delves into the complex topic of nonlinear electrophoresis of highly charged particles in electrolyte solutions, focusing on the impact of ion diffusivity and correlation effects on particle mobility. A modified electrokinetic model is introduced to analyze these phenomena, showing overscreening and mobility reversal in multivalent electrolytes. The study compares the modified model with traditional approaches, highlighting the importance of considering short-range effects in understanding electrophoresis behavior. Additionally, the document includes a list of related scientific research articles covering various aspects of electrophoresis and electrokinetic phenomena, offering valuable insights into the behavior of charged particles in electric fields. [Extracted from the article]

In this paper, we study the following quasilinear Schrödinger equation: - ε 2 Δ u + V (x) u - ε 2 Δ (u 2) u = g (u) , x ∈ R 2 , where ε > 0 is a small parameter, V ∈ C (R 2 , R) is uniformly positive and allowed to be unbounded from above, and g ∈ C (R , R) has a critical exponential growth at infinity. In the autonomous case, when ε > 0 is fixed and V (x) ≡ V 0 ∈ R + , we first present a remarkable relationship between the existence of least energy solutions and the range of V 0 without any monotonicity conditions on g. Based on some new strategies, we establish the existence and concentration of positive solutions for the above singularly perturbed problem. In particular, our approach not only permits to extend the previous results to a wider class of potentials V and source terms g, but also allows a uniform treatment of two kinds of representative nonlinearities that g has extra restrictions at infinity or near the origin, namely lim inf | t | → + ∞ t g (t) e α 0 t 4 or g (u) ≥ C q , V u q - 1 with q > 4 and C q , V > 0 is an implicit value depending on q, V and the best constant of the embedding H 1 (R 2) ⊂ L q (R 2) , considered in the existing literature. To the best of our knowledge, there have not been established any similar results, even for simpler semilinear Schrödinger equations. We believe that our approach could be adopted and modified to treat more general elliptic partial differential equations involving critical exponential growth. [ABSTRACT FROM AUTHOR]

We studied an obstacle problem of a $ p $-Laplacian type in a bounded non-smooth domain whose nonlinearity is associated with a matrix weight. We established a global Calderón-Zygmund theory by proving that the gradient of a bounded weak solution is as integrable as both the gradient of the obstacle and the nonhomogeneous term in the weighted Lebesgue space with the desired estimate under possibly minimal regularity requirements of the given operator, which are a small log-BMO of the matrix weight, a weak small BMO in $ x $ variable and local uniform continuity in the solution $ u $ variable of the nonlinearity $ {\bf a}(x, u, Du) $, and a $ \delta $-Reifenberg flatness of the boundary. Our regularity theory on the obstacle problem covered a larger class of elliptic partial differential equations in the literature by allowing degenerate/singular nonlinearities, also depending on the solution $ u $. [ABSTRACT FROM AUTHOR]

The paper is devoted to the Dirichlet problem in a flat bounded domain for a linear secondorder functional differential equation in the divergent form with dilation, contraction and rotation of the argument of the higher-order derivatives of the unknown function. We study the existence, the uniqueness and the smoothness of the generalized solution for all possible values of the coefficients and parameters of transformations in the equation. [ABSTRACT FROM AUTHOR]

A class of semilinear elliptic differential equations was investigated in this study. By constructing the inverse function, using the method of upper and lower solutions and the principle of comparison, the existence of the maximum positive solution and the minimum positive solution was explored. Furthermore, the uniqueness of the positive solution and its asymptotic estimation at the origin were evaluated. The results show that the asymptotic estimation is similar to that of the corresponding boundary blowup problems. Compared with the conclusions of Wei's work in 2017, the asymptotic behavior of the solution only depends on the asymptotic behavior of b (x) at the origin and the asymptotic behavior of g at infinity. [ABSTRACT FROM AUTHOR]

The Journal of Fluid Mechanics article introduces a mathematical model for wind-generated particle-fluid flow fields, specifically addressing the helicopter cloud problem. The model examines the interaction between fluid and suspended particles above a static particle bed, focusing on entrainment and detrainment processes. By enhancing predictability in scenarios like the helicopter cloud problem, the research sheds light on the formation of fluidized clouds in diverse applications, such as desert and oceanic particle transport phenomena. [Extracted from the article]

The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze the convergence properties of discretizations and obtain, in two dimensions, an a priori error estimate for the underlying approximation of an optimal control variable. [ABSTRACT FROM AUTHOR]

In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss because of the mismatch between the true and assumed models. A robustness problem in this context is to show that the error because of the mismatch between a true and an assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we discuss continuity and robustness properties of finite and infinite horizon α-discounted/ergodic optimal control problems for a general class of nondegenerate controlled diffusion processes as well as for optimal control up to an exit time. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error because of the mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We see that, compared with related results in the discrete-time setup, the continuous-time theory lets us utilize the strong regularity properties of solutions to optimality (Hamilton–Jacobi–Bellman) equations, via the theory of uniformly elliptic partial differential equations, to arrive at strong continuity and robustness properties. Funding: The research of S. Yüksel was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). [ABSTRACT FROM AUTHOR]

Using the Pohozaev-type arguments, we prove the nonexistence results for sub-elliptic problems with critical Sobolev-Hardy exponents and Hardy-type potentials on the Carnot group. [ABSTRACT FROM AUTHOR]

In the regularity theory of solutions to elliptic partial differential equations often the concept of stability plays the role of a sufficient condition for smoothness. It is a natural question to ask if this holds true for nonstable but finite Morse index solutions. We provide a negative answer showing the existence of sequences of solutions with radial Morse index equal to 1 for which regularity estimates can not be satisfied. [ABSTRACT FROM AUTHOR]

The document "In memoriam Alfred K Louis" published in the journal "Inverse Problems" pays tribute to the late Professor Alfred K Louis, a distinguished mathematician known for his work in inverse problems. It highlights his significant contributions to mathematics, particularly in tomography and the Radon transform, as well as his impact as a mentor and collaborator. The article details his academic journey, research achievements, and professional legacy, emphasizing his dedication to students and the field of applied mathematics. The text also acknowledges his numerous awards and honors, underscoring his profound influence on the scientific community. [Extracted from the article]

Our main goal is to investigate supercritical Hardy–Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term. [ABSTRACT FROM AUTHOR]

The article "A reciprocal theorem for biphasic poro-viscoelastic materials" in the Journal of Fluid Mechanics presents a reciprocal theorem for biphasic materials with a solid phase and a fluid phase, allowing for the calculation of integrated quantities like net force without solving additional problems. The study examines the forces on a rigid sphere in response to point forces applied to the solid and fluid phases, revealing time-dependent force evolution. The reciprocal theorem has practical applications in slip calculations, nonlinear effects, and self-propulsion in biphasic systems, offering insights into complex fluid dynamics in porous media. [Extracted from the article]

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions. [ABSTRACT FROM AUTHOR]

In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BV one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BV regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases. [ABSTRACT FROM AUTHOR]

In a sequence of papers, the author examined several statistical affinity measures for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (Cnodes) in algebraic multigrid (AMG) for systems of elliptic partial differential equations (PDEs). These measures were applied to a set of relaxed vectors that exposes the problematic error components. Once the CDOFs are determined using any one of these measures, the interpolation operator is constructed in a bootstrap AMG (BAMG) procedure. However, in a recent paper of Kahl and Rottmann, the statistical least angle regression (LARS) method was utilized in the coarsening procedure and shown to be promising in the CDOF selection. This method is generally used in the statistics community to select the most relevant variables in constructing a parsimonious model for a very complicated and high‐dimensional model or data set (i.e., variable selection for a "reduced" model). As pointed out by Kahl and Rottmann, the LARS procedure has the ability to detect group relations between variables, which can be more useful than binary relations that are derived from strength‐of‐connection, or affinity measures, between pairs of variables. Moreover, by using an updated Cholesky factorization approach in the regression computation, the LARS procedure can be performed efficiently even when the original set of variables is large; and due to the LARS formulation itself (i.e., its l1$$ {l}_1 $$‐norm constraint), sparse interpolation operators can be generated. In this article, we extend the LARS coarsening approach to systems of PDEs. Furthermore, we incorporate some modifications to the LARS approach based on the so‐called elastic net and relaxed lasso methods, which are well known and thoroughly analyzed in the statistics community for ameliorating several major issues with LARS as a variable selection procedure. We note that the original LARS coarsening approach may have addressed some of these issues in similar or other ways but due to the limited details provided there, it is difficult to determine the extent of their similarities. Incorporating these modifications (or effecting them in similar ways) leads to improved robustness in the LARS coarsening procedure, and numerical experiments indicate that the changes lead to faster convergence in the multigrid method. Moreover, the relaxed lasso modification permits an indirect BAMG (iBAMG) extension to the interpolation operator. This iBAMG extension applied in an intra‐ or inter‐variable interpolation setting (i.e., nodal‐based coarsening), as well as in variable‐based coarsening, which will not preserve the nodal structure of a finest‐level discretization on the lower levels of the multilevel hierarchy, will be examined. For the variable‐based coarsening, because of the parsimonious feature of LARS, the performance is reasonably good when applied to systems of PDEs albeit at a substantial additional cost over a nodal‐based procedure. [ABSTRACT FROM AUTHOR]

In this paper we propose a weaker version of Penrose's much heeded Strong Cosmic Censorship (SCC) conjecture, asserting inextendability of maximal Cauchy developments by manifolds with Lipschitz continuous Lorentzian metrics and Riemann curvature bounded in Lp. Lipschitz continuity is the threshold regularity for causal structures, while curvature bounds rule out infinite tidal accelerations, arguing for physical significance of this weaker SCC conjecture. The main result of this paper, under the assumption that no extensions exist with higher connection regularity W loc 1 , p , proves in the affirmative this SCC conjecture with bounded curvature for p sufficiently large, (p > 4 to address uniform bounds, p > 2 without uniform bounds). [ABSTRACT FROM AUTHOR]

This work is on a fast and accurate reduced basis method for solving discretized fractional elliptic partial differential equations (PDEs) of the form Asu = f by rational approximation. A direct computation of the action of such an approximation would require solving multiple (20~30) large-scale sparse linear systems. Our method constructs the reduced basis using the first few directions obtained from the preconditioned conjugate gradient method applied to one of the linear systems. As shown in the theory and experiments, only a small number of directions (5~10) are needed to approximately solve all large-scale systems on the reduced basis subspace. This reduces the computational cost dramatically because: (1) We only use one of the large-scale problems to construct the basis; and (2) all large-scale problems restricted to the subspace have much smaller sizes. We test our algorithms for fractional PDEs on a 3d Euclidean domain, a 2d surface, and random combinatorial graphs. We also use a novel approach to construct the rational approximation for the fractional power function by the orthogonal greedy algorithm (OGA) [ABSTRACT FROM AUTHOR]

The fractional differential equation L^\beta u = f posed on a compact metric graph is considered, where \beta >0 and L = \kappa ^2 - \nabla (a\nabla) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients \kappa,a. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L^{-\beta }. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L_2(\Gamma \times \Gamma)-error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for {L = \kappa ^2 - \Delta, \kappa >0} are performed to illustrate the results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the smoothness of generalized solutions of the first boundary-value problem for a strongly elliptic functional differential equation containing orthotropic contraction transformations of the arguments of the unknown function in the leading part. The problem is considered in a disc, the coefficients of the equation are constant. Orthotropic contraction is understood as different contraction in different variables. Conditions for the conservation of smoothness on the boundaries of neighboring subdomains formed by the action of the contraction transformation group onto the disc are found in explicit form for any right-hand side from the Lebesgue space. [ABSTRACT FROM AUTHOR]

In the framework of Clifford analysis, we study a second‐order elliptic (generally nonstrongly elliptic) system of partial differential equations of the form: ν∂x_fϑ∂x_=0$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}}{f}^{\vartheta}\kern-0.1em {\partial}_{\underset{\_}{x}}=0 $$, where ν∂x_$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}} $$ stands for the Dirac operator with respect to a structural set ν$$ \nu $$. The solutions of this system are known as (ν,ϑ)$$ \left(\nu, \vartheta \right) $$‐inframonogenic functions. Our main purpose is to describe necessary and sufficient conditions for the solvability of a ∂¯$$ \overline{\partial} $$‐problem associated with the sandwich operator ν∂x_(·)ϑ∂x_$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}}{\left(\cdotp \right)}^{\vartheta}\kern-0.1em {\partial}_{\underset{\_}{x}} $$. [ABSTRACT FROM AUTHOR]

Algebraic multigrid (AMG) methods are among the most efficient solvers for linear systems of equations and they are widely used for the solution of problems stemming from the discretization of Partial Differential Equations (PDEs). A severe limitation of AMG methods is the dependence on parameters that require to be fine-tuned. In particular, the strong threshold parameter is the most relevant since it stands at the basis of the construction of successively coarser grids needed by the AMG methods. We introduce a novel deep learning algorithm that minimizes the computational cost of the AMG method when used as a finite element solver. We show that our algorithm requires minimal changes to any existing code. The proposed Artificial Neural Network (ANN) tunes the value of the strong threshold parameter by interpreting the sparse matrix of the linear system as a gray scale image and exploiting a pooling operator to transform it into a small multi-channel image. We experimentally prove that the pooling successfully reduces the computational cost of processing a large sparse matrix and preserves the features needed for the regression task at hand. We train the proposed algorithm on a large dataset containing problems with a strongly heterogeneous diffusion coefficient defined in different three-dimensional geometries and discretized with unstructured grids and linear elasticity problems with a strongly heterogeneous Young's modulus. When tested on problems with coefficients or geometries not present in the training dataset, our approach reduces the computational time by up to 30%. [ABSTRACT FROM AUTHOR]

We derive an extension of the sequential homotopy method that allows for the application of inexact solvers for the linear (double) saddle‐point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off‐diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method allows for the parallel solution of large 3D problems. [ABSTRACT FROM AUTHOR]

We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs remains limited in the literature. To address this gap, we establish a proof of global convergence of these new iterative algorithms using a discrete comparison principle argument. We also provide a specific implementation for the Monge-Ampére equation. Several numerical tests are performed to validate the convergence theorem. These numerical experiments involve examples of varying regularity. Computational experiments show that method is efficient, robust, and requires relatively few iterations to converge. The results reveal great potential for DDM methods to lead to highly efficient and parallelizable solvers for large-scale problems that are computationally intractable using existing solution methods. [ABSTRACT FROM AUTHOR]

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form. [ABSTRACT FROM AUTHOR]

We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods. [ABSTRACT FROM AUTHOR]

AbstractWe prove the uniformity of the in-plane and anti-plane elastic field of stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity embedded in an infinite linear isotropic elastic matrix subjected to uniform remote in-plane and anti-plane stresses. The elastic material occupying the elliptical inhomogeneity obeys Neuber’s special nonlinear stress-strain law. The original boundary value problem is finally reduced to a single non-linear equation for the constant effective strain within the inhomogeneity, which is rigorously proved to have a unique solution. Once the non-linear equation is solved numerically, we establish the uniform elastic field within the elliptical inhomogeneity and the non-uniform elastic field in the linear elastic matrix. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For the spatial discretization of elliptic and parabolic partial differential equations (PDEs), we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order k ∈ N. On a quite general class of 2D domains, namely separable domains , and even on special surfaces, the discrete problem is then reformulated as a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. By considering the IMEX Euler method for the PDE time discretization, we obtain a sequence of these equations. To solve each multiterm Sylvester equation, we apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method with a matrix-oriented preconditioner that captures the spectral properties of the Sylvester operator. Solving the Poisson problem and the heat equation on some separable domains by MO-FEM-PCG, we show a gain in computational time and memory occupation wrt the classical vector PCG with same preconditioning or wrt a LU based direct method. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns on some separable domains and cylindrical surfaces for a morphochemical reaction-diffusion PDE system for battery modelling. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence and uniqueness of weak solution for the problem...is bounded and open. Here, the functions f: O R and g: O × R R satisfy the suitable hypotheses. [ABSTRACT FROM AUTHOR]

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, to guarantee well-posedness, are known [(see, e.g., Exercise 2.14 of Ern and Guermond (Theory and practice of finite elements, Applied mathematical, sciences, Springer, 2004)]. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately. [ABSTRACT FROM AUTHOR]

Spatial network models are used as a simplified discrete representation in a wide range of applications, e.g., flow in blood vessels, elasticity of fiber based materials, and pore network models of porous materials. Nevertheless, the resulting linear systems are typically large and poorly conditioned and their numerical solution is challenging. This paper proposes a numerical homogenization technique for spatial network models which is based on the super-localized orthogonal decomposition (SLOD), recently introduced for elliptic multiscale partial differential equations. It provides accurate coarse solution spaces with approximation properties independent of the smoothness of the material data. A unique selling point of the SLOD is that it constructs an almost local basis of these coarse spaces, requiring less computations on the fine scale and achieving improved sparsity on the coarse scale compared to other state-of-the-art methods. We provide an a posteriori analysis of the proposed method and numerically confirm the method's unique localization properties. In addition, we show its applicability also for high-contrast channeled material data. [ABSTRACT FROM AUTHOR]

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 s -Gevrey assumptions on the residual equation, we establish s -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. [ABSTRACT FROM AUTHOR]

A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations—such as coefficient identification in partial differential equations (PDEs) from boundary observations—can often be achieved by extending the searched for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton-type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs. [ABSTRACT FROM AUTHOR]

We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e. the total computational time. Numerical experiments underline the theory. [ABSTRACT FROM AUTHOR]

In this work, we establish existence results for the following fourth-order Kirchhoff-type elliptic problem with Hardy potential M(∫Ω |∆u| pdx) ∆²pu - a |x|p|u|p-2u = λf(x, u), in Ω, u = ∆u = 0, on ∂Ω. Precisely, by using the classical Hardy inequality and critical point theory, we prove the existence of multiple weak solutions for the fourth-order Kirchhoff-type elliptic problem with Hardy potential. [ABSTRACT FROM AUTHOR]

For any compact connected one-dimensional submanifold |$K\subset \mathbb R^{2\times 2}$| without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$ This is an optimal generalization, for compact connected submanifolds of |$\mathbb R^{2\times 2}$| without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into |$SO(n)$|⁠. The proof relies on the special properties of elliptic subsets |$K\subset{{\mathbb{R}}}^{2\times 2}$| with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion |$Du\in K$|⁠. We also give an example showing that no analogous result can hold true in |$\mathbb R^{n\times n}$| for |$n\geq 3$|⁠. [ABSTRACT FROM AUTHOR]

We describe a compact finite-difference discretization and Gaussian-radial basis function for the two-dimensional local fractional elliptic PDEs that describe anomalous diffusion-convection of groundwater contamination. Precisely estimating pollutant concentration over a long period helps protect water reservoirs. The local fractional partial differential equations and their discretization described here are the generalization of the integer order elliptic partial differential equations and their high-order scheme. The high-order discretization of fractal gradient and anomalous diffusion on a non-uniformly spaced nine-point single-cell grid network gives the result in small computing time. The new scheme is supported by a detailed convergence analysis describing the monotone property and a strongly connected Jacobian (iteration) matrix graph. The computational illustration of various anomalous diffusion-convection models demonstrates the proposed methodology's effectiveness. [ABSTRACT FROM AUTHOR]

For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem. [ABSTRACT FROM AUTHOR]

In a recent paper, the author examined a correlation affinity measure for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (C nodes) in systems of elliptic partial differential equations (PDEs). This measure was applied to a set of relaxed vectors, which exposed the near‐nullspace components of the PDE operator. Selecting the CDOFs using this affinity measure and constructing the interpolation operators using a least‐squares procedure, an algebraic multigrid (AMG) method was developed. However, there are several noted issues with this AMG solver. First, to capture strong anisotropies, a large number of test vectors may be needed; and second, the solver's performance can be sensitive to the initial set of random test vectors. Both issues reflect the sensitive statistical nature of the measure. In this article, we derive several other statistical measures that ameliorate these issues and lead to better AMG performance. These measures are related to a Markov process, which the PDE itself may model. Specifically, the measures are based on the diffusion distance/effective resistance for such process, and hence, these measures incorporate physics into the CDOF selection. Moreover, because the diffusion distance/effective resistance can be used to analyze graph networks, these measures also provide a very economical scheme for analyzing large‐scale networks. In this article, the derivations of these measures are given, and numerical experiments for analyzing networks and for AMG performance on weighted‐graph Laplacians and systems of elliptic boundary‐value problems are presented. [ABSTRACT FROM AUTHOR]

In an open, bounded Lipschitz polygon Ω ⊂ R 2 , we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in Ω . The boundary conditions on each edge of ∂ Ω are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in H 1 (Ω) , quantized tensor formats and certain deep neural networks. [ABSTRACT FROM AUTHOR]

The 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 G$G$‐convergence arguments. [ABSTRACT FROM AUTHOR]