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We introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree $p$. In the setting of [S. Bertoluzza and D. Prada, A polygonal discontinuous Galerkin method with minus one stabilization, ESAIM Math. Mod. Numer. Anal. (DOI: 10.1051/m2an/2020059)], the stabilization is obtained by penalizing, in each mesh element $K$, a residual in the norm of the dual of $H^1(K)$. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a $p$-explicit stability and error analysis, proving $p$-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments., 31 pages, 3 figures, 9 tables

The shifted boundary method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods. It has proven to be quite efficient in handling problems with complex geometries, ranging from Poisson to Darcy, from Navier-Stokes to elasticity and beyond. The key feature of the SBM is a shift in the location where Dirichlet boundary conditions are applied—from the true to a surrogate boundary—and an appropriate modification (again, a shift) of the value of the boundary conditions, in order to reduce the consistency error. In this paper we provide a sound analysis of the method in smooth domains and in domains with corners, highlighting the influence of geometry and distance between exact and surrogate boundaries upon the convergence rate. We consider the Poisson problem with Dirichlet boundary conditions as a model and we first detail a procedure to obtain the crucial shifting between the surrogate and the true boundaries. Next, we give a sufficient condition for the well-posedness and stability of the discrete problem. The behavior of the consistency error arising from shifting the boundary conditions is thoroughly analyzed, for smooth boundaries and for boundaries with corners and edges. The convergence rate is proven to be optimal in the energy norm, and is further enhanced in the L 2 L^2 -norm.

We investigate the piecewise linear nonconforming Crouzeix–Raviart and the lowest order Raviart–Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix–Raviart and the Raviart–Thomas finite-element approximate problems. We next present the equivalence between the Raviart–Thomas finite-element method and the enriched Crouzeix–Raviart finite-element method. We emphasize that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained.

We study the Schwarz overlapping domain decomposition method applied to the Poisson problem on a special family of domains, which by construction consist of a union of a large number of fixed-size subdomains. These domains are motivated by applications in computational chemistry where the subdomains consist of van der Waals balls. As is usual in the theory of domain decomposition methods, the rate of convergence of the Schwarz method is related to a stable subspace decomposition. We derive such a stable decomposition for this family of domains and analyze how the stability "constant" depends on relevant geometric properties of the domain. For this, we introduce new descriptors that are used to formalize the geometry for the family of domains. We show how, for an increasing number of subdomains, the rate of convergence of the Schwarz method depends on specific local geometry descriptors and on one global geometry descriptor. The analysis also naturally provides lower bounds in terms of the descriptors for the smallest eigenvalue of the Laplace eigenvalue problem for this family of domains., Comment: published in SIAM Journal on Numerical Analysis

A non-intrusive extension to the standard p -version of the finite element method is proposed. Meshes with hanging nodes are handled by adapting the reference elements so that the resulting discretisation is always conforming. The shape functions on these adaptive reference elements are not polynomials, but either harmonic extensions of the boundary restrictions of the standard shape functions or solutions to a local Poisson problem. The numerical experiments are taken from computational function theory and the efficiency of the proposed extension resulting in exponential convergence in the quantities of interest is demonstrated.

We provide an elementary derivation of the Green function for Poisson’s equation with Neumann boundary data on balls of arbitrary dimension. Surprisingly, until very recently this Green function wa...

International audience In this work, we consider conforming finite element discretizations of arbitrary polynomial degree $p \ge 1$ of the Poisson problem. We propose a multilevel a posteriori estimator of the algebraic error. We prove that this estimator is reliable and efficient (represents a two-sided bound of the error), with a constant independent of the degree $p$. We next design a multilevel iterative algebraic solver from our estimator and show that this solver contracts the algebraic error on each iteration by a factor bounded independently of $p$. Actually, we show that these two results are equivalent. The $p$-robustness results rely on the work of Schöberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1–24] for one given mesh. We combine this with the design of an algebraic residual lifting constructed over a hierarchy of nested unstructured, possibly highly graded, simplicial meshes. The lifting includes a global coarse-level solve with the lowest polynomial degree one together with local contributions from the subsequent mesh levels. These contributions, of the highest polynomial degree $p$ on the finest mesh, are given as solutions of mutually independent local Dirichlet problems posed over overlapping patches of elements around vertices. The construction of this lifting can be seen as one geometric V-cycle multigrid step with zero pre- and one post-smoothing by (damped) additive Schwarz (block Jacobi). One particular feature of our approach is the optimal choice of the step-size generated from the algebraic residual lifting. Numerical tests are presented to illustrate the theoretical findings.

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt class. Second, we prove the stability in weighted norms for standard finite element approximations under the quasi-uniformity assumption on the family of meshes., 13 pages

The purpose of this paper is to study and classify singular solutions of the Poisson problem { L μ s u = f in Ω ∖ { 0 } , u = 0 in R N ∖ Ω \begin{equation*} \left \{ \begin {aligned} \mathcal {L}^s_\mu u = f \quad \ \text {in}\ \, \Omega \setminus \{0\},\\ u =0 \quad \ \text {in}\ \, \mathbb {R}^N \setminus \Omega \ \end{aligned} \right . \end{equation*} for the fractional Hardy operator L μ s u = ( − Δ ) s u + μ | x | 2 s u \mathcal {L}_\mu ^s u= (-\Delta )^s u +\frac {\mu }{|x|^{2s}}u in a bounded domain Ω ⊂ R N \Omega \subset \mathbb {R}^N ( N ≥ 2 N \ge 2 ) containing the origin. Here ( − Δ ) s (-\Delta )^s , s ∈ ( 0 , 1 ) s\in (0,1) , is the fractional Laplacian of order 2 s 2s , and μ ≥ μ 0 \mu \ge \mu _0 , where μ 0 = − 2 2 s Γ 2 ( N + 2 s 4 ) Γ 2 ( N − 2 s 4 ) > 0 \mu _0 = -2^{2s}\frac {\Gamma ^2(\frac {N+2s}4)}{\Gamma ^2(\frac {N-2s}{4})}>0 is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case μ = μ 0 \mu = \mu _0 which requires more subtle estimates than the case μ > μ 0 \mu >\mu _0 .

A new finite element method/boundary element method (FEM/BEM) scheme is proposed for the solution of the 2D magnetic static and quasi-static problems with unbounded domains. The novelty is an original approach in the treatment of the outer region. The related domain integral is eliminated at the discrete level by using the finite element approximation of the fundamental solutions (Green’s functions) at every node of the related mesh. This “FEM-Green” approach replaces the standard boundary element method. It is simpler to implement because no integration on the boundary of the domain is required. Then, the method leads to a substantially reduced computational burden. Moreover, the coupling with finite elements is more natural since it is based on the same Galerkin approximation. Some examples with open boundary and nonlinear materials are presented and compared with the standard finite element method.

In this note we are concerned with interior regularity properties of the $p$-Poisson problem $��_p(u)=f$ with $p>2$. For all $0, 25 pages

We prove norm-resolvent and spectral convergence in $L^2$ of solutions to the Neumann Poisson problem $-\Delta u_\varepsilon = f$ on a domain $\Omega_\varepsilon$ perforated by Dirichlet-holes and shrinking to a 1-dimensional interval. The limit $u$ satisfies an equation of the type $-u''+\mu u = f$ on the interval $(0,1)$, where $\mu$ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighbourhood and the vertex neighbourhood is chosen correctly, the constant $\mu$ will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation., Comment: 21 pages, 3 figures

We present a unifying viewpoint on hybrid high-order and virtual element methods on general polytopal meshes in dimension $2$ or $3$, in terms of both formulation and analysis. We focus on a model Poisson problem. To build our bridge (i) we transcribe the (conforming) virtual element method into the hybrid high-order framework and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local virtual element discrete bilinear form is defined. This allows us to perform a unified analysis of virtual element/hybrid high-order methods, that differs from standard virtual element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

A refined a priori error analysis of the lowest order (linear) nonconforming Virtual Element Method (VEM) for approximating a model Poisson problem is developed in both 2D and 3D. A set of new geometric assumptions is proposed on shape regularity of polytopal meshes. A new error equation for the lowest order (linear) nonconforming VEM is derived for any choice of stabilization, and a new stabilization using a projection on an extended element patch is introduced for the error analysis on anisotropic elements.

The notion of the core inverse of tensors with the Einstein product was introduced, very recently (Sahoo et al. Comp Appl Math 39:9, 2020). In this paper we establish a few sufficient and necessary conditions for reverse-order law of this inverse. Further, we discuss mixed-type reverse-order law for core inverse. In addition to these, we discuss core inverse solutions of multilinear systems via the Einstein product. The prowess of the inverse is demonstrated to solve the Poisson problem in the multilinear system framework.

We treat three problems on a two-dimensional “punctured periodic domain”: we take [Formula: see text], where [Formula: see text] and [Formula: see text] is the closure of an open connected set that is star-shaped with respect to [Formula: see text] and has a [Formula: see text] boundary. We impose periodic boundary conditions on the boundary of [Formula: see text], and Dirichlet boundary conditions on [Formula: see text]. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier–Stokes equations, all with a fixed forcing function [Formula: see text], and examine the behavior of solutions as [Formula: see text]. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of [Formula: see text] with periodic boundary conditions.

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampere equation with Dirichlet boundary conditions. We formulate the Monge-Ampere equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.

We consider a model Poisson problem in $\R^d$ ($d=2,3$) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges ($d=2$) or small faces ($d=3$)., Comment: 36 pages

International audience; The ghost-cell immersed boundary methods (IBMs) are widely used to implement boundary conditions on non-body fitted grids. It has been previously shown that they have two major drawbacks. Firstly, these methods tend to have a maximum stencil size larger than 1, yielding non-band matrices. Secondly, for a maximum stencil size of 2 the ghost-cell linear IBM [1] have a first-order convergence rate for Neumann immersed boundary conditions. To address these two shortcomings and in the pursuit of increased accuracy and increased order of convergence the current article proposes the linear/quadratic square shifting methods for the ghost-cell IBM for Cartesian grids. The linear square shifting method guarantees a maximum stencil size of 1 and also improves the accuracy and convergence while the quadratic square shifting method improves the accuracy and convergence while maintaining the same stencil size of 2 as the original linear method [1]. The quadratic ghost-cell method [2], [3] further improves the accuracy and convergence whilst maintaining a maximum stencil size of 3 while the currently proposed quadratic square shifting method makes it possible to increase the Lagrange polynomial interpolation order whilst maintaining a maximum stencil of 2 resulting in an improved order of convergence for Neumann immersed boundary conditions. The proposed methods are evaluated by considering their accuracy and convergence thanks to a comprehensive verification and validation process. Firstly, the canonical verification 2D and 3D Poisson test problems for various analytical solutions and immersed boundaries are considered and are followed by various verification and validation test cases for the Navier-Stokes governing equations, with and without heat transfer, in 2D and 3D.

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete $H^1$ norm best approximation error estimates for $H^2$ functions hold for arbitrary triangulations. However, similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on the example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily slow speed. The results complement analogous findings for conforming P1 elements., 23 pages, 10 figures

In the paper, the asymptotic solutions for a problem of Cauchy–Poisson type with localized initial conditions are constructed. The bottom of the basin under consideration which was constant before the perturbation, is instantly perturbed at the initial time moment by a spatially localized function. Simplifications of the corresponding formulas are presented inside and outside the vicinity of the leading front, as well as in the case of a special choice of the initial condition. It is shown that, in the vicinity of the leading front, the asymptotic solution coincides with the asymptotic solution of the linear Boussinesq equation.

A residual-based a posteriori error estimate for the Poisson problem with discontinuous diffusivity coefficient is derived in the case of a virtual element discretization. The error is measured considering a suitable polynomial projection of the discrete solution to prove an equivalence between the defined error and a computable residual based error estimator that does not involve any term related to the virtual element stabilization. Numerical results display a very good behavior of the ratio between the error and the error estimator, resulting independent of the meshsize and element distortion.

The focus of this work is on the development of an error-driven isogeometric framework, capable of automatically performing an adaptive simulation in the context of second- and fourth-order, elliptic partial differential equations defined on two-dimensional trimmed domains. The method is steered by an a posteriori error estimator, which is computed with the aid of an auxiliary residual-like problem formulated onto a space spanned by splines with single element support. The local refinement of the basis is achieved thanks to the use of truncated hierarchical B-splines. We prove numerically the applicability of the proposed estimator to various engineering-relevant problems, namely the Poisson problem, linear elasticity and Kirchhoff–Love shells, formulated on trimmed geometries. In particular, we study several benchmark problems which exhibit both smooth and singular solutions, where we recover optimal asymptotic rates of convergence for the error measured in the energy norm and we observe a substantial increase in accuracy per-degree-of-freedom compared to uniform refinement. Lastly, we show the applicability of our framework to the adaptive shell analysis of an industrial-like trimmed geometry modeled in the commercial software Rhinoceros, which represents the B-pillar of a car.

Given a sequence of finite element spaces which form a de Rham sequence, we will construct a dual representation of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The matrix which converts primal representations to dual representations -- the Hodge matrix -- is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even for high order methods. The derivative of dual representations will be defined. Vector operations, grad, curl and div, for primal and dual representations are both topological and do not depend on the metric, i.e. the size and shape of the mesh or the order of the numerical method. Derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations. As illustration of the use of dual representations, the method will be applied to i) a mixed formulation for the Poisson problem in 3D, ii) it will be shown that this approach allows one to preserve the equivalence between Dirichlet and Neumann problems in the finite dimensional setting and, iii) the method will be applied to the approximation of grad-div eigenvalue problem on affine and non-affine meshes., Comment: This is version5 of this article

In this paper, for the first time in literature, we introduce one-sided weighted inverses and extend the notions of one-sided inverses, outer inverses and inverses along given elements. Although our results are new and in the matrix case, we decided to present them in tensor space with reshape operator. For this purpose, a left and right ( M,N ) -weighted ( B,C ) -inverse and the ( M,N ) -weighted ( B,C ) -inverse of a tensor are defined. Additionally, necessary and sufficient conditions for the existence of these new inverses are presented. We describe the sets of all left (or right) ( M,N ) -weighted ( B,C ) -inverses of a given tensor. As consequences of these results, we consider the one-sided ( B,C ) -inverse, ( B,C ) -inverse, one-sided inverse along a tensor and inverse along a tensor. Further, we introduce a ( M,N ) -weighted ( B , C ) -outer inverse and a W -weighted ( B , C ) -outer inverse of tensors with a few characterizations. Then, corresponding algorithms for computing various types of outer inverses of tensors are proposed, implemented and tested. The prowess of the proposed inverses are demonstrated for finding the solution of Poisson problem and the restoration of 3D color images.

International audience; We present a novel formulation for the mesh adaptation of the approximation of a Partial Differential Equation (PDE). The discussion is restricted to a Poisson problem. The proposed norm-oriented formulation extends the goal-oriented formulation since it is equation-based and uses an adjoint. At the same time, the norm-oriented formulation somewhat supersedes the goal-oriented one since it is basically a solution-convergent method. Indeed, goal-oriented methods rely on the reduction of the error in evaluating a chosen scalar output with the consequence that, as mesh size is increased (more degrees of freedom), only this output is proven to tend to its continuous analog while the solution field itself may not converge. A remarkable quality of goal-oriented metric-based adaptation is the mathematical formulation of the mesh adaptation problem under the form of the optimization, in the well-identified set of metrics, of a well-defined functional. In the new proposed formulation, we amplify this advantage. We search, in the same well-identified set of metrics, the minimum of a norm of the approximation error. The norm is prescribed by the user and the method allows addressing the case of multi-objective adaptation like, for example in aerodynamics, adaptating the mesh for drag, lift and moment in one shot. In this work, we consider the basic linear finite-element approximation and restrict our study to L 2 norm in order to enjoy second-order convergence. Numerical examples for the Poisson problem are computed.

International audience; We define a reiterated unfolding operator for a doubly periodic domain presenting two periodicity scales. Then we show how to apply it to the homogenization of both linear and nonlinear problems. The main novelty is that this method allows the use of test functions with one scale of periodicity only and it considerably simplifies the proofs of the convergence results. We illustrate this new approach on a Poisson problem with Dirichlet boundary conditions and on the flow of a power law fluid in a doubly periodic porous medium

We introduce a novel technique—lean complementarity—that attempts to eliminate any waste of computational resources occurring during the pursuing of complementarity. First, contrarily to the widely used practice of solving the problem two times with a pair of complementary or complementary-dual formulations, lean complementarity requires just one solution with the computationally cheap formulation based on the scalar potential. This result is enabled by a novel and explicit flux equilibration technique that produces tight bounds and is computationally inexpensive, because no system has to be solved. Second, the systems arising during the adaptive mesh refinement procedure are solved inexactly on purpose, by stopping the iterations of the iterative solver when the algebraic error gets negligible with respect to the discretization error. The discretization error is bounded with complementarity, whereas the algebraic error is computed very accurately with a novel and cheap technique.

We show that, on a $2$-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the $L^2$-norm by identity plus the gradient of the solution to the Poisson problem $-\Delta f^{n,t} = \mu^{n,t}-1$, where $\mu^{n,t}$ is an appropriate regularization of the empirical measure associated to the random points. This shows that the ansatz of Caracciolo et al. (Scaling hypothesis for the Euclidean bipartite matching problem) is strong enough to capture the behavior of the optimal map in addition to the value of the optimal matching cost. As part of our strategy, we prove a new stability result for the optimal transport map on a compact manifold., Comment: 20 pages

We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming approximations of the Poisson problem to nonconforming Crouzeix-Raviart approximations of the Poisson and the Stokes problem in 2D. As a consequence, we obtain instance optimality of an AFEM with a modified maximum marking strategy.

We consider an Extended Finite Elements method to handle the case of composite independent unstructured grids that lead to unfitted meshes. In particular, we address the case of two overlapped meshes, a background and a foreground one, where the thickness of the latter is smaller than the elements of the background mesh. This situation may lead to elements split into several portions, thus generating polyhedral elements. We detail the corresponding discrete formulation for the Poisson problem with discontinuous coefficients. We also provide some technical details for the 3D implementation. Finally, we provide some numerical examples with the aim of showing the effectiveness of the proposed formulation.

International audience; We propose and validate a novel extension of Hybrid High-Order (HHO) methods to meshes featuring curved elements. HHO methods are based on discrete unknowns that are broken polynomials on the mesh and its skeleton. We propose here the use of physical frame polynomials over mesh elements and reference frame polynomials over mesh faces. With this choice, the degree of face unknowns must be suitably selected in order to recover on curved meshes analogous convergence rates as on straight meshes. We provide an estimate of the optimal face polynomial degree depending on the element polynomial degree and on the so-called effective mapping order. The estimate is numerically validated through specifically crafted numerical tests. We also give effective practical indications on how to choose the face polynomial degree on more standard problems closer to real life configurations, all corroborated by numerical evidence. In the worst case scenario of dealing with randomly distorted mesh sequences it is typically sufficient to select polynomial spaces over faces only one degree higher than on the elements to obtain optimal convergence rates. All test cases are conducted considering two-and three-dimensional pure diffusion problems, and include comparisons with DG discretizations. The extension to agglomerated meshes with curved boundaries is also considered.

A refined a priori error analysis of the lowest order (linear) Virtual Element Method (VEM) is developed for approximating a model two dimensional Poisson problem. A set of new geometric assumptions is proposed on shape regularity of polygonal meshes. A new universal error equation for the lowest order (linear) VEM is derived for any choice of stabilization, and a new stabilization using broken half-seminorm is introduced to incorporate short edges naturally into the a priori error analysis on isotropic elements. The error analysis is then extended to a special class of anisotropic elements with high aspect ratio originating from a body-fitted mesh generator, which uses straight lines to cut a shape regular background mesh. Lastly, some commonly used tools for triangular elements are revisited for polygonal elements to give an in-depth view of these estimates' dependence on shapes.

In this paper we investigate the behavior of the condition number of the stiffness matrix resulting from the approximation of a 2D Poisson problem by means of the Virtual Element Method. It turns out that ill-conditioning appears when considering high-order methods or in presence of "bad-shaped" (for instance nonuniformly star-shaped, with small edges...) sequences of polygons. We show that in order to improve such condition number one can modify the definition of the internal moments by choosing proper polynomial functions that are not the standard monomials. We also give numerical evidence that, at least for a 2D problem, standard choices for the stabilization give similar results in terms of condition number., Comment: 20 pages, 13 figures

The fractional nonlocal linearized Monge–Ampere equation is introduced. A Harnack inequality for nonnegative solutions to the Poisson problem on Monge–Ampere sections is proved.

International audience; We present a numerical method named Mixed High Order (MHO) to obtain high order of convergence for electrostatic problems solved on general polyhedral meshes. The method, based on high-order local reconstructions of differential operators from face and cell degrees of freedom, exhibits a moderate computational cost thanks to hybridization and static condensation that eliminate cell unknowns. After surveying the method, we first assess its effectiveness for three-dimensional problems by comparing for the first time its performances with classical conforming finite elements. Moreover, we emphasize the algebraic equivalence of MHO in the lowest-order with the analog formulation obtained with the Discrete Geometric Approach or the Finite Integration Technique.

In this paper, we present a unified analysis of both convergence and optimality of adaptive mixed finite element methods for a class of problems when the finite element spaces and corresponding a posteriori error estimates under consideration satisfy five hypotheses. We prove that these five conditions are sufficient for convergence and optimality of the adaptive algorithms under consideration. The main ingredient for the analysis is a new method to analyze both discrete reliability and quasi-orthogonality. This new method arises from an appropriate and natural choice of the norms for both the discrete displacement and stress spaces, namely, a mesh-dependent discrete $H^1$ norm for the former and a $L^2$ norm for the latter, and a newly defined projection operator from the discrete stress space on the coarser mesh onto the discrete divergence free space on the finer mesh. As applications, we prove these five hypotheses for the Raviart--Thomas and Brezzi--Douglas--Marini elements of the Poisson and Stokes problems in both 2D and 3D.

We consider the normalized $p$-Poisson problem $$-��^N_p u=f \qquad \text{in}\quad ��.$$ The normalized $p$-Laplacian $��_p^{N}u:=|D u|^{2-p}��_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,��}_{loc}$ regularity with nearly optimal $��$ for viscosity solutions of this problem. In the case $f\in L^{\infty}\cap C$ and $p>1$ we use methods both from viscosity and weak theory, whereas in the case $f\in L^q\cap C$, $q>\max(n,\frac p2,2)$, and $p>2$ we rely on the tools of nonlinear potential theory., 41 pages

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux and scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect.

We discuss a technique for the approximation of the Poisson problem under mixed boundary conditions in spaces of piece-wise constant functions. The method adopts ideas from the theory of $\Gamma$-convergence as a guideline. Some applications are considered and numerical evaluation of the convergence rate is discussed.

This document has been prepared to highlight a correction for the manuscript “A Matlab-Based Finite Difference Solver for the Poisson Problem with Mixed Dirichlet–Neumann Boundary Conditions”.

We prove L∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and p-Laplacian, namely $$\begin{array}{ll} -\Delta_p^N u=f\quad{\rm for } \; n < p \leq\infty.\end{array}$$ We are able to provide a stable family of results depending continuously on the parameter p. We also prove the failure of the classical Alexandrov–Bakelman–Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.

International audience; This note establishes regularity estimates for the solution of the Maxwell equations in Lipschitz domains with non-smooth coe cients and minimal regularity assumptions. The argumentation relies on elliptic regularity estimates for the Poisson problem with non-smooth coe cients.

Fully computable, guaranteed bounds are obtained on the error in the finite element approximation which take the effect of the boundary approximation into account. We consider the case of piecewise affine approximation of the Poisson problem with pure Neumann boundary data, and obtain a fully computable quantity which is shown to provide a guaranteed upper bound on the energy norm of the error. The estimator provides, up to a constant and oscillation terms, local lower bounds on the energy norm of the error.

International audience; We formulate and solve the Poisson problem for the exterior derivative operator with Dirichlet boundary condition in Lipschitz domains, of arbitrary topology, for data in Besov and Triebel-Lizorkin spaces.