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Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L 2 - and H 1 -norms are derived for the problem with initial data u 0 ∈ H 0 1 (Ω) ∩ H 2 (Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H 1 -norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L ∞ error estimate is obtained for 2D problems that too with the initial condition is in H 0 1 (Ω) ∩ H 2 (Ω). All results proved in this paper are valid uniformly as α → 1 − , where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, we investigate a divergence-free, nonconforming virtual element method for the numerical approximation of the Stokes eigenvalue problems using polygonal unstructured meshes on polygonal domain. By employing an elliptic projection operator, we "enhance" the definition of the virtual element space to make the L 2 -projection operator computable with optimal order. Furthermore, we prove the optimal order of convergence of the eigenfunction approximation and the double order of convergence of the eigenvalue approximation through compactness arguments for the solution operator (Babuška-Osborn Theory). Finally, we present a set of numerical experiments to assess the good approximation behavior of the method and show its computational performance. [ABSTRACT FROM AUTHOR]

In this paper we consider the vector-valued Schrödinger operator − Δ + V , where the potential term V is a matrix-valued function whose entries belong to L loc 1 (R d) and, for every x ∈ R d , V (x) is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in L 1 (R d , R m). Assuming further that the minimal eigenvalue of V belongs to some reverse Hölder class of order q ∈ (1 , ∞) ∪ { ∞ } , we obtain maximal inequality in L p (R d , R m) , for p in between 1 and some q , and generation results. [ABSTRACT FROM AUTHOR]

In this paper, elliptic control problems with the integral constraint on the gradient of the state and the box constraint on the control are considered. The optimality conditions for the problem are proved. To numerically solve the problem, a finite element duality-based inexact majorized accelerated block coordinate descent (FE-dABCD) algorithm is proposed. Specifically, both the state and the control are discretized by piecewise linear functions. An inexact majorized ABCD algorithm is employed to solve the discretized problem via its dual, which is a multi-block unconstrained convex optimization problem, but the primal variables are also generated in each iteration. Thanks to the inexactness of the FE-dABCD algorithm, the subproblems at each iteration are allowed to be solved inexactly. For the smooth subproblem, we use the preconditioned generalized minimal residual (GMRES) method to solve it. For the two nonsmooth subproblems, one of them has a closed form solution through introducing an appropriate proximal term, and another one is solved by the line search Newton's method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys O (1 k 2 ) iteration complexity. Moreover, to make the algorithm more efficient and further reduce its computation cost, based on the mesh-independence of ABCD method, we propose an FE-dABCD algorithm with a warm-start strategy (wFE-dABCD). Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm and wFE-dABCD algorithm. [ABSTRACT FROM AUTHOR]

This paper studies a priori and regularity estimates of Evans-Krylov type in Hölder spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems. [ABSTRACT FROM AUTHOR]

In this paper we propose a variant of enriched Galerkin methods for second order elliptic equations with over-penalization of interior jump terms. The bilinear form with interior over-penalization gives a non-standard norm which is different from the discrete energy norm in the classical discontinuous Galerkin methods. Nonetheless we prove that optimal a priori error estimates with the standard discrete energy norm can be obtained by combining a priori and a posteriori error analysis techniques. We also show that the interior over-penalization is advantageous for constructing preconditioners robust to mesh refinement by analyzing spectral equivalence of bilinear forms. Numerical results are included to illustrate the convergence and preconditioning results. • A new locally mass conservative finite element method with interior over-penalized enriched Galerkin methods (IOP-EG). • A priori error analysis of IOP-EG independent of over-penalization. • A robust preconditioner construction and analysis by operator preconditioning approach. [ABSTRACT FROM AUTHOR]

In this study, a new Poisson-type equation for non-hydrostatic pressure in terrain-following σ -coordinates is proposed. The resulting equation consists of a standard elliptic operator and other first-order operators for non-hydrostatic pressure. Compared with the original Poisson-type equation, the treatment of the boundary condition at the lateral boundary can be simplified. Numerical tests indicate that the reformulated Poisson-type equation shows better convergence properties than the original equation when the quadrature-free nodal discontinuous Galerkin method is adopted for numerical discretization. Additionally, for the standing wave case, the numerical error of the three-dimensional non-hydrostatic model with the reformulated Poisson equation is smaller than that with the original equation, the numerical results of the solitary wave case indicate that the dissipation of the three-dimensional non-hydrostatic model is small, and transformation of wave over a circular shoal indicates that the developed model can properly simulate combined wave refraction and diffraction. [ABSTRACT FROM AUTHOR]

We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case, we define a new functional whose critical points are proved to be precisely the Kähler metrics, if any, on the manifold. As general manifolds of either type need not admit Kähler metrics, this provides an approach to new obstructions to Kählerianity within these two families of metrics. [ABSTRACT FROM AUTHOR]

For a real analytic family of almost complex structures (M , J t) , t ∈ T , on an even dimensional compact real analytic manifold M , we prove upper semi-continuity with respect to t of the dimension of the space of Δ ∂ ‾ t -harmonic (p , q) -forms in Zariski sense. More general statements in terms of the dimension of the kernel of a family of self-adjoint strongly elliptic operators on a vector bundle over a manifold in real analytic settings are proved. [ABSTRACT FROM AUTHOR]

In this paper, we solve a second-order nonlinear elliptic equation by using the finite volume element method, and give the rigorous error estimates. Firstly, the computational domain is divided into general convex quadrilateral meshes. We choose the isoparametric bilinear element space as the trial function space and the piecewise constant function space as the test function space, and construct the corresponding finite volume element scheme. Secondly, on the h 2 -parallelogram mesh, the boundedness and coercivity of bilinear form are proved. Using the Brouwer fixed point theorem, we give the existence and uniqueness of numerical solution. Thirdly, we derive the estimates of ‖ ∇ (u − u h) ‖ L t with t ≥ 2 and ‖ u − u h ‖ 0 under certain regularity assumptions. At last, we carry out numerical experiments on quadrilateral meshes and calculate the convergence orders in H 1 and L 2 norms, which are consistent with our theoretical results. [ABSTRACT FROM AUTHOR]

When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for matrix-sequences arising from the approximation of the Laplacian via ad hoc finite differences. The analysis involves several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix-sequences. Several numerical experiments are conducted, which confirm the correctness of the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness and H 2 -regularity of solution u for a semi-linear elliptical equation with a 2 m − 1 polynomial nonlinearity under the stability and uniqueness conditions. Moreover, we design the finite element method based on P 1 -element for the semi-linear elliptical equation and prove the existence and uniqueness of the finite element solution u h by the finite element Oseen iterative method under the stability and uniqueness condition. Next, we prove the existence and uniqueness of the solution u for the semi-linear elliptical equation by the finite element solution u h under the stability and uniqueness conditions. Furthermore, we provide the H 1 − L 2 optimal error estimates of the finite element solution u h with respect to the exact solution u for the semi-linear elliptical equation under the stability and uniqueness conditions. Finally, some numerical tests are presented to show the efficiency of the proposed method for the semi-linear elliptical equation with a 2 m − 1 polynomial nonlinearity. [ABSTRACT FROM AUTHOR]

We establish new quantitative Hessian integrability estimates for viscosity supersolutions of fully nonlinear elliptic operators. As a corollary, we show that the optimal Hessian power integrability ε = ε (λ , Λ , n) in the celebrated W 2 , ε –regularity estimate satisfies (1 + 2 3 (1 − λ Λ)) n − 1 ln ⁡ n 4 ⋅ (λ Λ) n − 1 ≤ ε ≤ n λ (n − 1) Λ + λ , where n ≥ 3 is the dimension and 0 < λ < Λ are the ellipticity constants. In particular, (Λ λ) n − 1 ε (λ , Λ , n) blows-up, as n → ∞ ; previous results yielded fast decay of such a quantity. The upper estimate improves the one obtained by Armstrong, Silvestre, and Smart in [1]. [ABSTRACT FROM AUTHOR]

We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is C 1 , α on the boundary of the domain, for an optimal and explicit α given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, and independent of the elliptic operator. We also obtain sharp global estimates. We use a different method from all previous results of global nature, which give C 1 , α regularity only for some small α > 0. Our findings are new even for the model equations, involving a degenerate Laplacian. [ABSTRACT FROM AUTHOR]

We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by Alessandrini in 1990. Our solutions can be applied to the determination of the optical properties of an anisotropic medium in time-harmonic Diffuse Optical Tomography (DOT). [ABSTRACT FROM AUTHOR]

The approximation properties of shifted fundamental solutions are stated for a general class of second order linear elliptic differential operators. Motivated by the method of the fundamental solutions, the singularities of the fundamental solutions are located in the vicinity of a fixed surface, which is the boundary of a 3-dimensional Lipschitz domain. The approximation is estimated first in C k -norm. The corresponding convergence rate is determined depending on the density of the singularities. This delivers a solid theoretical basis for the method of fundamental solutions. The results are extended to exterior domains and as a new application, the numerical solution of Dirichlet-to-Neumann problems are constructed and analyzed on unbounded domains. The efficiency of this new method is demonstrated in series of numerical experiments. [ABSTRACT FROM AUTHOR]

We study the insulated conductivity problem with inclusions embedded in a bounded domain in R n , for n ≥ 3. The gradient of solutions may blow up as ε , the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on S n − 2. [ABSTRACT FROM AUTHOR]

In this paper, we consider the uniformly elliptic nonlocal Bellman problem F s u (x) = f (u (x) , v (x)) , F s v (x) = g (u (x) , v (x)). Firstly, we study narrow region principles for the uniformly elliptic nonlocal Bellman operators in bounded and unbounded domains, which play key roles in obtaining the main results by the process of sliding method. Then we deal with monotonicity properties of solutions to the uniformly elliptic nonlocal Bellman system. [ABSTRACT FROM AUTHOR]

On a smooth, closed Riemannian manifold (M , g) of dimension n ≥ 3 , we consider the stationary Schrödinger equation Δ g u + h 0 u = | u | 2 ⁎ − 2 u , where Δ g : = − div g ∇ , h 0 ∈ C 1 (M) and 2 ⁎ : = 2 n n − 2. We prove that, up to perturbations of the potential function h 0 in C 1 (M) , the sets of sign-changing solutions that are bounded in H 1 (M) are precompact in the C 2 topology. We obtain this result under the assumptions that (M , g) is locally conformally flat, n ≥ 7 and h 0 ≠ n − 2 4 (n − 1) Scal g at all points in M , where Scal g is the scalar curvature of the manifold. We then provide counterexamples in every dimension n ≥ 3 showing the optimality of these assumptions. [ABSTRACT FROM AUTHOR]

We study the link between Ψdos and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of Ψdos of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in terms of anti-Wick operators which leads to a sharp Gårding inequality and transition of hypoellipticity between Wick and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators. [ABSTRACT FROM AUTHOR]

Motivated by a challenging expectation of Rivière [24] , in the recent interesting work [5] , de Longueville and Gastel proposed the following geometrical even order elliptic system Δ m u = ∑ l = 0 m − 1 Δ l 〈 V l , d u 〉 + ∑ l = 0 m − 2 Δ l δ (w l d u) in B 2 m which includes polyharmonic mappings as special cases. Under minimal regularity assumptions on the coefficient functions and an additional algebraic antisymmetry assumption on the first order potential, they successfully established a conservation law for this system, from which everywhere continuity of weak solutions follows. This beautiful result amounts to a significant advance in the expectation of Rivière. In this paper, we seek for the optimal interior regularity of the above system, aiming at a more complete solution to the aforementioned expectation of Rivière. Combining their conservation law and some new ideas together, we obtain optimal Hölder continuity and sharp L p regularity theory, similar to that of Sharp and Topping [27] , for weak solutions to a related inhomogeneous system. Our results can be applied to study heat flow and bubbling analysis for polyharmonic mappings. [ABSTRACT FROM AUTHOR]

In this paper, a new form of two-point nonlocal boundary conditions is presented. This condition generalizes a Dirichlet condition at one boundary and a mixed condition at the other one. The focus of this work is not the numerical solution of the considered problem. Rather, we study the effect of the proposed nonlocal boundary condition on the difference eigenvalue problem for an elliptic partial differential equation in one and two dimensions. [ABSTRACT FROM AUTHOR]

Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in applications. The universal approximation properties of INNs have been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems. [ABSTRACT FROM AUTHOR]

In this paper, the novel localized singular boundary method (LSBM) in conjunction with the Chebyshev collocation scheme (CCS) is developed to simulate large-scale inhomogeneous problems governed by elliptic operators. By means of the Gauss–Lobatto collocation points, the CCS is employed to obtain the particular solution. With the obtained particular solution, the inhomogeneous elliptic boundary value problem can be transformed into a classical homogeneous problem which can be solved by the LSBM. In the LSBM, the whole computational domain with distributed nodes is divided into a series of overlapping circle-shaped sub-domains. For each sub-domain, the moving least square method and the conventional singular boundary method (SBM) are utilized. Considering that the matrix generated by the CCS-LSBM is sparse, this scheme is granted the tremendous potential to address large-scale problems. Numerical experiments are presented to verify the effectiveness and accuracy of the proposed scheme for problems with mixed boundary conditions, sophisticated analytical solutions, and complex computational domains, respectively. [ABSTRACT FROM AUTHOR]

We define the fractional powers L s = (− a i j (x) ∂ i j) s , 0 < s < 1 , of nondivergence form elliptic operators L = − a i j (x) ∂ i j in bounded domains Ω ⊂ R n , under minimal regularity assumptions on the coefficients a i j (x) and on the boundary ∂Ω. We show that these fractional operators appear in several applications such as fractional Monge–Ampère equations, elasticity, and finance. The solution u to the nonlocal Poisson problem { (− a i j (x) ∂ i j) s u = f in Ω u = 0 on ∂ Ω is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge–Ampère geometry and prove the interior Harnack inequality and Hölder estimates for solutions to the extension problem when the coefficients a i j (x) are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions u to the fractional problem. [ABSTRACT FROM AUTHOR]

We propose, analyze mathematically, and study numerically a novel approach for the finite element approximation of the spectrum of second-order elliptic operators. The main idea is to reduce the stiffness of the problem by subtracting a least-squares penalty on the gradient jumps across the mesh interfaces from the standard stiffness bilinear form. This penalty bilinear form is similar to the known technique used to stabilize finite element approximations in various contexts. The penalty term is designed to dampen the high frequencies in the spectrum and so it is weighted here by a negative coefficient. The resulting approximation technique is called softFEM since it reduces the stiffness of the problem. The two key advantages of softFEM over the standard Galerkin FEM are to improve the approximation of the eigenvalues in the upper part of the discrete spectrum and to reduce the condition number of the stiffness matrix. We derive a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form. Then we prove that softFEM delivers the same optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. We next compare the discrete eigenvalues obtained when using Galerkin FEM and softFEM. Finally, a detailed analysis of linear softFEM for the 1D Laplace eigenvalue problem delivers a sensible choice for the softness parameter. With this choice, the stiffness reduction ratio scales linearly with the polynomial degree. Various numerical experiments illustrate the benefits of using softFEM over Galerkin FEM. [ABSTRACT FROM AUTHOR]

Let H be the first Heisenberg group, and let k ∈ C ∞ (H ∖ { 0 }) be a kernel which is either odd or horizontally odd, and satisfies | ∇ H n k (p) | ≤ C n ‖ p ‖ − 1 − n , p ∈ H ∖ { 0 } , n ≥ 0. The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel k (p) = ∇ H log ⁡ ‖ p ‖. We prove that convolution with k , as above, yields an L 2 -bounded operator on regular curves in H. This extends a theorem of G. David to the Heisenberg group. As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield L 2 bounded operators on Lipschitz flags in H. This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li. [ABSTRACT FROM AUTHOR]

In this paper, adaptive direct discontinuous Galerkin method for second order elliptic equations is studied in two dimensional. A numerical flux with general weighted averages and proper weights for interface problems is introduced. In addition to the optimal a priori error estimator, we propose a residual-type a posteriori error estimator. The global upper bound and local lower bound for the error in DG norm are established. Several numerical examples are carried out to confirm the reliability and efficiency of the proposed error estimator and the corresponding adaptive direct discontinuous Galerkin method. [ABSTRACT FROM AUTHOR]

In this paper, the inverse problem of the determination of an unknown discontinuous parameter in elliptic equations is considered. A constrained optimization problem is formulated using the TV regularization and a specified selection term, in order to absorb the effect of noise and singularities. The discretization scheme used in this work is a combined finite element/volume method, then we establish the discrete version of optimization problem. The convergence and stability analysis are performed in term of noise level and diameter of singular set. We have considered the projected Wolfe algorithm to design our proposed minimization method. At the end we show some numerical experiments to prove the validity of theoretical results and the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

For Itô stochastic processes in R d with drift in L d Aleksandrov's type estimates are established in the elliptic and parabolic settings. They are applied to estimating the resolvent operators of the corresponding elliptic and parabolic operators in L p and L p + 1 , respectively, where p ≥ d. [ABSTRACT FROM AUTHOR]

Let M be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a torus T , and P 0 an invariant elliptic classical pseudodifferential operator on M. In this note, we strengthen the asymptotics for the equivariant (or reduced) spectral function of P 0 derived in [6] , which are already sharp in the eigenvalue aspect, to become almost sharp in the isotypic aspect. In particular, this leads to hybrid equivariant L p -bounds for eigenfunctions that are almost sharp in the eigenvalue and isotypic aspect. [ABSTRACT FROM AUTHOR]

The purpose of this work is to show the iterate property for globally elliptic differential operators with polynomial coefficients (called Shubin operators), in the anisotropic Roumieu Gelfand-Shilov spaces S { N } { M } (R n). As a consequence we obtain a result of regularity of solutions of differential equations in these spaces. [ABSTRACT FROM AUTHOR]

The propagation of internal gravity waves in stratified media, such as those found in ocean basins and lakes, leads to the development of geometrical patterns called "attractors". These structures accumulate much of the wave energy and make the fluid flow highly singular. In more analytical terms, the cause of this phenomenon has been attributed to the presence of a continuous spectrum in some nonlocal zeroth-order pseudo-differential operators. In this work, we analyse the generation of these attractors from a numerical analysis perspective. First, we propose a high-order pseudo-spectral method to solve the evolution problem (whose long-term behaviour is known to be not square-integrable). Then, we use similar tools to discretize the corresponding eigenvalue problem. Since the eigenvalues are embedded in a continuous spectrum, we compute them using viscous approximations. Finally, we explore the effect that the embedded eigenmodes have in the long-term evolution of the system. [ABSTRACT FROM AUTHOR]

We present the universal one-loop effective action up to dimension eight after integrating out heavy fermion(s) using the Heat-Kernel method. We have discussed how the Dirac operator being a weak elliptic operator, the fermionic operator still can be written in the form of a strong elliptic one such that the Heat-Kernel coefficients can be used to compute the fermionic effective action. This action captures the footprint of both the CP conserving as well as violating UV interactions. As it does not rely on the specific forms of either UV or low energy theories, can be applicable for a very generic action. Our result encapsulates the effects of heavy fermion loop through Yukawa couplings only. [ABSTRACT FROM AUTHOR]

In this work we determine the second-order coefficient in a parabolic équation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate of the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of stability estimate is based on a spectral decomposition of the solution to the parabolic equation in terms of the eigenfunctions of the associated elliptic operator, and an ad hoc method to solve a nonlinear stationary transport equation that is itself of interest. [ABSTRACT FROM AUTHOR]

We propose to solve elliptic interface problems by a meshless finite difference method, where the second order elliptic operator and jump conditions are discretized with the help of the QR decomposition of an appropriately rescaled multivariate Vandermonde matrix with partial pivoting. A prescribed consistency order is achieved on irregular nodes with small influence sets, which allows to place the nodes directly on the unfitted interface and leads to sparse system matrices with the density of nonzero entries comparable to the density of the system matrices arising from the mesh-based finite difference or finite element methods. Numerical experiments on a number of standard test problems with known solutions demonstrate convergence orders up to O (h 6) for both the approximate solution and its gradient, and a robust performance of the method in the case when the interface is known inaccurately. [ABSTRACT FROM AUTHOR]

This paper presents some computer-assisted procedures to prove the invertibility of a second-order linear elliptic operator and to compute a bound for the norm of its inverse. These approaches are based on constructive L 2 -norm estimates of the Laplacian and improve on previous procedures that use projection and a priori error estimations. Several examples which confirm the actual effectiveness of the procedures are reported. [ABSTRACT FROM AUTHOR]

In this paper we develop and investigate numerical algorithms for solving equations of the first order for the fractional powers of discrete elliptic operators: A α y = φ , 0 < α < 1 , for φ ∈ V h with V h a finite element or discrete approximation space. The main goal is to consider two different approaches to construct efficient high order time stepping schemes for the implementation of the Cauchy problem method (Vabishchevich, 2015). The first approach is based on a solution of the related time-dependent PDE, and the second approach is based on diagonal Padé approximations to (1 + x) − α when the relation of the solution on two times levels is used. The second and fourth order approximations are constructed by both approaches. In order to increase the accuracy of approximations the graded time grid is constructed which compensates the singular behavior of the solution for t close to 0. Results of numerical experiments are presented, they agree well with the theoretical results. [ABSTRACT FROM AUTHOR]

A time-fractional initial–boundary value problem is considered, where the spatial domain has dimension d ∈ { 1 , 2 , 3 } , the spatial differential operator is a standard elliptic operator, and the time derivative is a Caputo derivative of order α ∈ (0 , 1). To discretise in space we use a standard piecewise-polynomial finite element method, while for the temporal discretisation the GMMP scheme (a variant of the Grünwald-Letnikov scheme) is used on a uniform mesh. The analysis of the GMMP scheme for solutions that exhibit a typical weak singularity at the initial time t = 0 has not previously been considered in the literature. A global convergence result is derived in L ∞ (L 2) , then a more delicate analysis of the error in this norm shows that, away from t = 0 , the method attains optimal-rate convergence. Numerical results confirm the sharpness of the theoretical error bounds. [ABSTRACT FROM AUTHOR]

We construct and study the weak solution to stochastic differential equation d X (t) = − b (X (t)) d t + 2 d W (t) , X (0) = x , for every x ∈ R d , d ≥ 3 , with b in the class of weakly form-bounded vector fields, containing, as proper subclasses, a sub-critical class [ L d + L ∞ ] d , as well as critical classes such as weak L d class, Kato class, Campanato–Morrey class. [ABSTRACT FROM AUTHOR]

We discuss the p - and h p -versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schrödinger equation with a pseudo-potential term. As an interesting byproduct, we present for the first time in literature an explicit construction of the stabilization of the mass matrix. We present in detail the analysis of the p -version of the method, proving exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing h p -refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the h p -spaces. [ABSTRACT FROM AUTHOR]

In this article, we prove Gevrey estimates of the resolvent near threshold zero for a class of second order elliptic operators satisfying a weighted coercive condition. As application, we obtain large time expansions with sub-exponential time-decay estimates on the remainder for the heat and Schrödinger semigroups generated by non-selfadjoint Schrödinger operators. Our results cover the cases of threshold eigenvalue and positive resonances. [ABSTRACT FROM AUTHOR]

This paper deals with the homogenization problem for convolution type non-local operators in random statistically homogeneous ergodic media. Assuming that the convolution kernel has a finite second moment and satisfies certain symmetry and uniform ellipticity conditions, we prove the almost sure homogenization result and show that the limit operator is a second order elliptic differential operator with constant deterministic coefficients [ABSTRACT FROM AUTHOR]

In this paper, we present an immersed weak Galerkin method for solving second-order elliptic interface problems. The proposed method does not require solution meshes to be aligned with the interface. Consequently, uniform Cartesian meshes can be used for nontrivial interfacial geometry. We show the existence and uniqueness of the numerical algorithm, and provide the error analysis in the energy norm. Numerical results are reported to demonstrate the performance of the method. [ABSTRACT FROM AUTHOR]

In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form ∇ ⋅ C → (x , ∇ u (x)) = 0 , in a bounded smooth domain Ω. We assume that C ⃗ (x , p →) = γ (x) p → + b → (x) | p → | 2 + O ( | p → | 3) , by expanding C ⃗ (x , p →) around p → = 0. We give a reconstruction method for γ and b → from the Dirichlet to Neumann map defined on ∂ Ω. [ABSTRACT FROM AUTHOR]

In this paper, we present a posteriori error estimate of weak Galerkin (WG) finite element methods based on the second order elliptic interface problems. This estimate can be applied to polygonal meshes or meshes with hanging nodes. The reliability and efficiency of the designed error estimator has been proved in this work. Extensive numerical tests are performed to validate our algorithm. These results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator. [ABSTRACT FROM AUTHOR]

Recently some neural networks have been proposed for computing approximate solutions to partial differential equations. For some second order elliptic or parabolic PDEs, error estimates are proved between the solution and the computed one with neural networks, assuming this one minimizes a L p norm or a dual norm of the residual, or an abstract loss function. In this article, for some second order elliptic PDEs, thanks to a gradient flow strategy, we prove the existence of a neural network solution minimizing the loss function with respect to the neural network parameters and we give an error estimate between the solution and the computed one with neural networks. For some nonsymmetric elliptic PDEs, the problem is expressed in form of a MinMax problem which is approximating with a double NN. Thanks to a diagonal extraction process a result of convergence is established with respect to the parameters of NNs and errors estimates are also given. [ABSTRACT FROM AUTHOR]

We study modular differential equations (MDEs) of the elliptic genus of four-dimensional complex varieties with trivial first Chern class. We construct modular differential equations of orders 3, 4, 5 and 6 with respect to the heat operator for every weak Jacobi form of weight 0 and index 2. We prove that the elliptic genus of a Calabi–Yau fourfold satisfies a MDE of the minimal possible order 3 if and only if its Euler number is equal to 48 or −18. We construct MDEs of order 5 of the elliptic genus of hyperkähler fourfolds of types Hilb2(K3) and Kum2(A). [ABSTRACT FROM AUTHOR]