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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]

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]

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]

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]

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we consider, from a numerical point of view, a two-temperature poro-thermoelastic problem. The model is written as a coupled linear system of hyperbolic and elliptic partial differential equations. An existence result is proved and energy decay properties are recalled. Then we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained, from which the linear convergence of the approximation is deduced under an appropriate additional regularity. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation, the decay of the discrete energy and the behaviour of the solution depending on a constitutive parameter. [ABSTRACT FROM AUTHOR]

In this article, we give a description of a technique to develop an a posteriori error estimator for the dual mixed methods, when applied to elliptic partial differential equations with non homogeneous mixed boundary conditions. The approach considers conforming finite elements for the discrete scheme, and a quasi‐Helmholtz decomposition result to obtain a residual a posteriori error estimator. After applying first a homogenization technique (for the Neumann boundary condition), we derive an a posteriori error estimator, which looks to be expensive to compute. This motivates the derivation of another a posteriori error estimator, that is fully computable. As a consequence, we establish the equivalence between the latter a posteriori error estimator and the natural norm of the error, that is, we prove the reliability and local efficiency of the aforementioned estimator. Finally, we report numerical examples showing the good properties of the estimator, in agreement with the theoretical results of this work. [ABSTRACT FROM AUTHOR]

It is well known that the standard discretization methods usually suffer from several numerical instabilities when simulating advection-dominated flows, requiring some enhancements. Therefore, stabilized formulations and/or adaptive mesh strategies are needed to suppress nonphysical oscillations. Beyond the stabilized formulations, additional special techniques, e.g., shock-capturing operators, are also required for resolving sharp layers accurately. For these reasons, asymptotic methods can be preferred as an alternative to numerical schemes for simulating such phenomena, especially when the behavior of solutions within and outside the boundary/inner layers is of interest. To that end, in this computational work, we adopt an asymptotic approach called the successive complementary expansion method (SCEM) and are interested in boundary layer solutions of elliptic partial differential equations. The regularly perturbed boundary-value problems arising from the SCEM process are discretized with the classical Galerkin finite element method. In this regard, we propose an asymptotic-numerical hybrid scheme. The results obtained reveal that the proposed method works quite well for moderately small values of the diffusion (perturbation) parameter. Besides, it achieves that even at the zeroth-order approximation. On the other hand, it is also observed that, depending on the performance of the complementary numerical scheme, the hybridized method needs additional treatment for challenging values of the diffusion parameter. Therefore, we also discuss possible remedies in detail. [ABSTRACT FROM AUTHOR]

This paper suggests reduced-order modeling using the Galerkin proper orthogonal decomposition (POD) to find approximate solutions for parabolic partial differential equations. We first transform a parabolic partial differential equation to the frequency-dependent elliptic equations using the Fourier integral transform in time. Such a frequency-domain method enables efficiently implementing a parallel computation to approximate the solutions because the frequency-variable elliptic equations have independent frequencies. Then, we introduce reduced-order modeling to determine approximate solutions of the frequency-variable elliptic equations quickly. A set of snapshots consists of the finite element solutions of the frequency-variable elliptic equations with some selected frequencies. The solutions are approximated using the general basis of the high-dimensional finite element space in a Hilbert space. reduced-order modeling employs the Galerkin POD for the snapshot subspace spanned by a set of snapshots. An orthonormal basis for the snapshot space can be easily computed using the spectral decomposition of the correlation matrix of the snapshots. Additionally, using an appropriate low-order basis of the snapshot space allows approximating the solutions of the frequency-variable elliptic equations quickly, where the approximate solutions are used for the inverse Fourier transforms to determine the approximated solutions in the time variable. Several numerical tests based on the finite element method are presented to asses the efficient performances of the suggested approaches. [ABSTRACT FROM AUTHOR]

The purpose of this study is to investigate the role that a deep learning approach could play in computational mechanics. In this paper, a convolutional neural network technique based on modified loss function is proposed as a surrogate of the finite element method (FEM). Several surrogate-based physics-informed neural networks (PINNs) are developed to solve representative boundary value problems based on elliptic partial differential equations (PDEs). According to the authors' knowledge, the proposed method has been applied for the first time to solve boundary value problems with elliptic partial differential equations as the governing equations. The results of the proposed surrogate-based approach are in good agreement with those of the conventional FEM. It is found that modification of the loss function could improve the prediction accuracy of the neural network. It is demonstrated that to some extent, the deep learning approach could replace the conventional numerical method as a significant surrogate model. [ABSTRACT FROM AUTHOR]

We present a numerical implementation for the Virtual Element Method that incorporates high order spaces. We include all the required computations in order to assemble the stiffness and mass matrices, and right hand side. Convergence of the method is verified for different polygonal partitions. An specific mesh-free application that allows to approximate harmonic functions is discussed, which is based on high-order projections onto polynomial spaces of degree k; this approach only requires to solve a k(k − 1)/2 linear system, reducing significantly the number of operations compared to usual finite or virtual element methods, and can be modified for different virtual spaces and elliptic partial differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the development of weak Galerkin (WG) finite element methods for optimal control problems governed by second order elliptic partial differential equations. WG methods are advantageous over other existing methods in that the control variable along the domain boundary is a natural variable by design in the numerical scheme. A convergence theory was established for the numerical solutions in various Sobolev norms. Numerical results are presented to demonstrate the efficiency and accuracy of the new scheme. [ABSTRACT FROM AUTHOR]

Optimal control problems are a class of optimisation problems with partial differential equations as constraints. These problems arise in many application areas of science and engineering. The finite element method was used to transform the optimal control problems of an elliptic partial differential equation into a system of linear equations of saddle point form. The main focus of this paper is to characterise and exploit the structure of the coeficient matrix of the saddle point system to build an eficient numerical process. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The numerical solution of saddle point problems is a computational task since well known numerical schemes perform poorly if they are not properly preconditioned. The main task of this paper is to construct a preconditioner the mimic the structure of the system coeficient matrix to accelerate the convergence of the generalised minimal residual method. Explicit expression of the eigenvalue and eigenvectors for the preconditioned matrix are derived. The main outcome is to achieve optimal convergence results in a small number of iterations with respect to the decreasing mesh size h and the changes in fi the regularisation problem parameters. The numerical results demonstrate the effectiveness and performance of the proposed preconditioner compared to the other existing preconditioners and confirm theoretical results. [ABSTRACT FROM AUTHOR]

We present a new finite element method, called ϕ‐FEM, to solve numerically elliptic partial differential equations with natural (Neumann or Robin) boundary conditions using simple computational grids, not fitted to the boundary of the physical domain. The boundary data are taken into account using a level‐set function, which is a popular tool to deal with complicated or evolving domains. Our approach belongs to the family of fictitious domain methods (or immersed boundary methods) and is close to recent methods of CutFEM/XFEM type. Contrary to the latter, ϕ‐FEM does not need any nonstandard numerical integration on cut mesh elements or on the actual boundary, while assuring the optimal convergence orders with finite elements of any degree and providing reasonably well conditioned discrete problems. In the first version of ϕ‐FEM, only essential (Dirichlet) boundary conditions was considered. Here, to deal with natural boundary conditions, we introduce the gradient of the primary solution as an auxiliary variable. This is done only on the mesh cells cut by the boundary, so that the size of the numerical system is only slightly increased. We prove theoretically the optimal convergence of our scheme and a bound on the discrete problem conditioning, independent of the mesh cuts. The numerical experiments confirm these results. [ABSTRACT FROM AUTHOR]

In this article we develop the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for elliptic partial differential equations with inhomogeneous Dirichlet, Neumann, and Robin boundary conditions, and the high contrast property emerges from the coefficients of elliptic operators and Robin boundary conditions. By careful construction of multiscale bases of the CEM-GMsFEM, we introduce two operators \scrD m and \scrN m which are used to handle inhomogeneous Dirichlet and Neumann boundary values and are also proved to converge independently of contrast ratios as enlarging oversampling regions. We providean a priori error estimate and show that the number of oversampling layers is the key factor in controlling numerical errors. A series of experiments are conducted, and those results reflect the reliability of our methods even with high contrast ratios. [ABSTRACT FROM AUTHOR]

General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. In the present work, we consider a random elliptic partial differential equation (PDE) where the discontinuous subordinated Gaussian random fields occur in the diffusion coefficient. Problem specific multilevel Monte Carlo (MLMC) Finite Element methods are constructed to approximate the mean of the solution to the random elliptic PDE. We prove a-priori convergence of a standard MLMC estimator and a modified MLMC—control variate estimator and validate our results in various numerical examples. [ABSTRACT FROM AUTHOR]

As an extension to the well-established stationary elliptic partial differential equation (PDE) with random continuous coefficients we study a time-dependent advection-diffusion problem, where the coefficients may have random spatial discontinuities. In a subsurface flow model, the randomness in a parabolic equation may account for insufficient measurements or uncertain material procurement, while the discontinuities could represent transitions in heterogeneous media. Specifically, a scenario with coupled advection and diffusion coefficients that are modeled as sums of continuous random fields and discontinuous jump components are considered. The respective coefficient functions allow a very flexible modeling, however, they also complicate the analysis and numerical approximation of the corresponding random parabolic PDE. We show that the model problem is indeed well-posed under mild assumptions and show measurability of the pathwise solution. For the numerical approximation we employ a sample-adapted, pathwise discretization scheme based on a finite element approach. This semi-discrete method accounts for the discontinuities in each sample, but leads to stochastic, finite-dimensional approximation spaces. We ensure measurability of the semi-discrete solution, which in turn enables us to derive moments bounds on the mean-squared approximation error. By coupling this semi-discrete approach with suitable coefficient approximation and a stable time stepping, we obtain a fully discrete algorithm to solve the random parabolic PDE. We provide an overall error bound for this scheme and illustrate our results with several numerical experiments. [ABSTRACT FROM AUTHOR]

Numerical methods for elliptic partial differential equations usually lead to systems of linear equations with sparse, symmetric, and positive definite matrices. In many methods, these matrices can be obtained as sums of local symmetric positive semidefinite matrices. In this article, we use this assumption and introduce a method that provides guaranteed lower and upper bounds to all individual eigenvalues of the preconditioned matrices. We apply the method for preconditioners arising from the same discretization problem but with simplified coefficients. The method uses solely the data over the solution domain and local connections between the degrees of freedom defined by the discretization. [ABSTRACT FROM AUTHOR]

In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the twodimensional heat equation. [ABSTRACT FROM AUTHOR]

We consider an elliptic partial differential equation with a random diffusion parameter discretized by a stochastic collocation method in the parameter domain and a finite element method in the spatial domain. We prove for the first time convergence of a stochastic collocation algorithm which adaptively enriches the parameter space as well as refines the finite element meshes. [ABSTRACT FROM AUTHOR]

The finite element method is used to develop a computational algorithm for solving a limited class of problems on the bending of composite plates reinforced with systems of unidirectional high-strength fibers. It is assumed that a neutral plane exists in the region of the plate and behaves similarly to a flexible nondeformable membrane. The displacements of the plate in the longitudinal direction are linear in thickness. In the case of fiber composites with different elastic moduli in tension and compression, the neutral plane, generally speaking, does not coincide with the median plane. The problem of minimizing the elastic energy functional in accordance with the Lagrange variational principle yields a fourth-order elliptic differential equation for the deflection. The bending stiffnesses of the plate included in the coefficients of the equation are calculated with account for the fact that the elastic characteristics of the reinforcing fibers under tension and compression are significantly different. The numerical solution of the equation is obtained via the finite element method with the help of a Bell triangular element. The paper presents computational results for the bending of rectangular laminated plates in which the fibers are laid in different directions. [ABSTRACT FROM AUTHOR]

The stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we analyze a class of truncation preconditioners for SGFEM. Extending the idea of the mean-based preconditioner, these preconditioners capture additional significant components of the stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a model problem and assuming affine-parametric representation of the diffusion coefficient, we perform spectral analysis of the preconditioned matrices and establish optimality of truncation preconditioners with respect to SGFEM discretization parameters. Furthermore, we report the results of numerical experiments for model diffusion problems with affine and nonaffine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear systems) and compare truncation preconditioners with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product ones. [ABSTRACT FROM AUTHOR]

This paper aims at treating the Finite element method (FEM) for the discretization of elliptic Partial differential equation (PDE) and focuses on FEM for solving two-dimensional Laplace equation in a sub-set of a square domain.The idea is to use finite element spaces that are induced by triangulations of a square domain to discretize the two dimensional elliptic Laplace equation on the surface. Then the two numerical solutions obtained by FEM based on the number of finite elements are compared to check the accuracy of the developed scheme. The FEM MATLAB Programming is used for the solution of two dimensional Laplace equations. Results are then compared with the analytic solution to check the accuracy of the developed scheme. [ABSTRACT FROM AUTHOR]

We propose a new algorithm for adaptive finite element methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by the application of a smoother. Even though these intermediate solutions are far from the exact algebraic solutions, their a posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost. We provide a qualitative analysis of how the error propagates throughout the algorithm, and we present a series of numerical experiments that highlight the efficiency and the computational speedup of S-AFEM. [ABSTRACT FROM AUTHOR]

This paper is devoted to a study of nonlinear Robin-type boundary value problems of second-order elliptic partial differential equations. Specifically, nonlinear boundary conditions are modeled by hemivariational inequalities, and the coefficients of the governing equations have a small periodic oscillating structure. We prove that the model problem can be homogenized; i.e., the sequence of solutions weakly converges to a homogenized solution as the periodicity approaches zero. We derive an O(∊1/2)-order estimate in H¹-norm for first-order asymptotic expansions. In particular, we provide an optimal O(∊)-order estimate in L²-norm for linear Robin boundary value problems. We analyze the numerical error of the Lagrange finite element method for the model problem and design a numerical experiment to validate the theoretical predictions. [ABSTRACT FROM AUTHOR]

We consider a system of second-order nonlinear elliptic partial differential equations that models the equilibrium configurations of a two-dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin (dG) finite element methods are used to approximate the solutions of this nonlinear problem with nonhomogeneous Dirichlet boundary conditions. A discrete inf–sup condition demonstrates the stability of the dG discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the nonlinear problem. A priori error estimates in the energy and |$\mathbf{L}^2$| norms are derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of the Newton iterates along with complementary numerical experiments. [ABSTRACT FROM AUTHOR]

A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, and preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data. [ABSTRACT FROM AUTHOR]

This paper studies adaptive first-order system least-squares finite element methods (LSFEMs) for second-order elliptic partial differential equations in nondivergence form. Unlike the classical finite element methods which use weak formulations of PDEs that are not applicable for the nondivergence equation, the first-order least-squares formulations naturally have stable weak forms without using integration by parts, allow simple finite element approximation spaces, and have built-in a posteriori error estimators for adaptive mesh refinements. The nondivergence equation is first written as a system of first-order equations by introducing the gradient as a new variable. Then, two versions of least-squares finite element methods using simple C0 finite elements are developed in the paper. One is the L²-LSFEM which uses linear elements, and the other is the W-LSFEM with a mesh-dependent weight to ensure the optimal convergence. Under the assumption that the PDE has a unique solution, a priori and a posteriori error estimates are presented. With an extra assumption on the operator regularity, convergence in standard norms for the W-LSFEM is also discussed. L²-error estimates are derived for both formulations. Extensive numerical experiments for continuous, discontinuous, and even degenerate coefficients on smooth and singular solutions are performed to test the accuracy and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric PDE problem. The performance of the adaptive algorithm is tested numerically for model problems with two different non-affine parametric representations of the coefficient. [ABSTRACT FROM AUTHOR]

In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface systems. In particular, we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fixed deterministic surface using a prescribed stochastic parametrisation of the random domain. An abstract analysis of a finite element discretisation coupled with a Monte-Carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived. The results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk-surface system and the theoretical convergence rates are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

Summary: In this work, we derive an a priori error estimate of order h2|log(h)| for the finite element approximation of a sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box‐constrains on the control are considered and finitely many pointwise state‐constrains are imposed on specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation for the state equation. [ABSTRACT FROM AUTHOR]

Summary: In the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(−a(x)∇·), with a continuous and positive over Ω‾, Ω being an open and bounded subset of Rd, d≥1. For the numerical approximation, we consider the classical Pk Finite Elements, in the case of Friedrichs–Keller triangulations, leading, as usual, to sequences of matrices of increasing size. The new results concern the spectral analysis of the resulting matrix‐sequences in the direction of the global distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on Ω=(0,1)2 and we give a brief account in the more involved case of variable coefficients and more general domains. Tools are drawn from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz sequences. Numerical results are shown for a practical evidence of the theoretical findings. [ABSTRACT FROM AUTHOR]

Poisson's equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference (FD) and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here we derive spectral methods for solving Poisson's equation on a square, cylinder, solid sphere and cube that have optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom used to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of FD matrices, our solver exploits a separated spectra property that holds for our carefully designed spectral discretizations. Without parallelization we can solve Poisson's equation on a square with 100 million degrees of freedom in under 2 min on a standard laptop. [ABSTRACT FROM AUTHOR]

We develop and analyse a stabilization term for cut finite element approximations of an elliptic second-order partial differential equation on a surface embedded in |${\mathbb{R}}^d$|⁠. The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces, and provides control of the variation of the finite element solution on the active three-dimensional elements that intersect the surface. We show that the condition number of the stiffness matrix is |$O(h^{-2})$|⁠ , where |$h$| is the mesh parameter. The stabilization term works for linear as well as for higher-order elements and the derivation of its stabilizing properties is quite straightforward, which we illustrate by discussing the extension of the analysis to general |$n$| -dimensional smooth manifolds embedded in |${\mathbb{R}}^d$|⁠ , with codimension |$d-n$|⁠. We also state the properties of a general stabilization term that are sufficient to prove optimal scaling of the condition number and optimal error estimates in energy- and |$L^2$| -norm. We finally present numerical studies confirming our theoretical results. [ABSTRACT FROM AUTHOR]

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped ℙ 1 {\mathbb{P}_{1}} conforming and non-conforming finite element and on the hybrid finite volume method. [ABSTRACT FROM AUTHOR]

In this paper we present a new adaptive two-stage algorithm for solving elliptic partial differential equations via a radial basis function collocation method. Our adaptive meshless scheme is based at first on the use of a leave-one-out cross validation technique, and then on a residual subsampling method. Each of phases is characterized by different error indicators and refinement strategies. The combination of these computational approaches allows us to detect the areas that need to be refined, also including the chance to further add or remove adaptively any points. The resulting algorithm turns out to be flexible and effective through a good interaction between error indicators and refinement procedures. Several numerical experiments support our study by illustrating the performance of our two-stage scheme. Finally, the latter is also compared with an efficient adaptive finite element method. [ABSTRACT FROM AUTHOR]

In this paper, we study a parameter identification problem for elliptic partial differential equations. We reconstruct the coefficient with additional boundary measurements, including both Dirichlet and Neumann boundary conditions. To solve the problem, the coupled complex boundary method (CCBM), originally proposed in Cheng et al. [A novel coupled complex boundary method for solving inverse source problems, Inverse Probl. 30 (2014), p. 055002] is used. With CCBM, a complex boundary problem is introduced in such a way that the boundary conditions are coupled in a complex Robin boundary condition with a parameter τ. Using Tikhonov regularization, the coefficient is sought such that the imaginary part of the solution of the forward Robin boundary value problem vanishes in the problem domain, which brings advantages on robustness in reconstruction. Besides, the reconstruction is feasible even for very small regularization parameter through choosing the values of τ properly. Some theoretical analyses are given. Moreover, noise model is analysed and the finite element method is used for discretization. Numerical examples show the feasibility and stability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the multi-region discontinuous Galerkin composite finite element method (MRDGCFEM) with hp-adaptivity for the discretization of second-order elliptic partial differential equations with discontinuous coefficients. This method allows for the approximation of problems posed on computational domains where the jumps in the diffusion coefficient form a micro-structure. Standard numerical methods could be used for such problems but the computational effort may be extremely high. Small enough elements to represent the underlying pattern in the diffusion coefficient have to be used. In contrast, the dimension of the underlying MRDGCFE space is independent of the complexity of the diffusion coefficient pattern. The key idea is that the jumps in the diffusion coefficient are no longer resolved by the mesh where the problem is solved; instead, the finite element basis (or shape) functions are adapted to the diffusion pattern allowing for much coarser meshes. In this paper, we employ hp-adaptivity on a series of test cases highlighting the practical application of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a distributed optimal control problem governed by elliptic partial differential equations with L2-norm constraint on the state variable. Firstly, the control problem is approximated by hp spectral element methods, which combines the advantages of the finite element methods with spectral methods; then, the optimality conditions of continuous system and discrete system are presented, respectively. Next, hp a posteriori error estimates are derived for the coupled state and control approximation. In the end, a projection gradient iterative algorithm is given, which solves the optimal control problems efficiently. Numerical experiments are carried out to confirm that the numerical results are in good agreement with the theoretical results. [ABSTRACT FROM AUTHOR]

We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by repeatedly updating the mesh node positions. It is well known that such a procedure eventually may lead to a deterioration of mesh quality, or even an invalidation of the mesh, when interior nodes penetrate neighboring cells. We examine this phenomenon, which can be traced back to the ineptness of the discretized objective when considered over the space of mesh node positions. As a remedy, we propose a restriction in the admissible mesh deformations, inspired by the Hadamard structure theorem. First and second order methods are considered in this setting. Numerical results show that mesh degeneracy can be overcome, avoiding the need for remeshing or other strategies. FEniCS code for the proposed methods is available on GitHub. [ABSTRACT FROM AUTHOR]

Consider the variational form of the ordinary integro-differential equation (OIDE) − u ″ + u + ∫ 0 1 q (⋅ , y) u (y) dy = f on the unit interval I , subject to homogeneous Neumann boundary conditions. Here, f and q respectively belong to the unit ball of H r (I) and the ball of radius M 1 of H s (I 2) , where M 1 ∈ [ 0 , 1). For ε > 0 , we want to compute ε -approximations for this problem, measuring error in the H 1 (I) sense in the worst case setting. Assuming that standard information is admissible, we find that the n th minimal error is Θ (n − min ⁡ { r , s / 2 }) , so that the information ε -complexity is Θ (ε − 1 / min ⁡ { r , s / 2 }) ; moreover, finite element methods of degree max ⁡ { r , s } are minimal-error algorithms. We use a Picard method to approximate the solution of the resulting linear systems, since Gaussian elimination will be too expensive. We find that the total ε -complexity of the problem is at least Ω (ε − 1 / min ⁡ { r , s / 2 }) and at most O (ε − 1 / min ⁡ { r , s / 2 } ln ⁡ ε − 1) , the upper bound being attained by using O (ln ⁡ ε − 1) Picard iterations. [ABSTRACT FROM AUTHOR]

The numerical simulation of cardiac electrophysiology is a highly challenging problem in scientific computing. The Bidomain system is the most complete mathematical model of cardiac bioelectrical activity. It consists of an elliptic and a parabolic partial differential equation (PDE), of reaction–diffusion type, describing the spread of electrical excitation in the cardiac tissue. The two PDEs are coupled with a stiff system of ordinary differential equations (ODEs), representing ionic currents through the cardiac membrane. Developing efficient and scalable preconditioners for the linear systems arising from the discretization of such computationally challenging model is crucial in order to reduce the computational costs required by the numerical simulations of cardiac electrophysiology. In this work, focusing on the Bidomain system as a model problem, we have benchmarked two popular implementations of the Algebraic Multigrid (AMG) preconditioner embedded in the PETSc library and we have studied the performance on the calibration of specific parameters. We have conducted our analysis on modern HPC architectures, performing scalability tests on multi-core and multi-GPUs settings. The results have shown that, for our problem, although scalability is verified on CPUs, GPUs are the optimal choice, since they yield the best performance in terms of solution time. [ABSTRACT FROM AUTHOR]

• Advection-dispersion solved at once by mixed and discontinuous finite elements. • Comparisons with analytical solutions show the accuracy of the new scheme. • Numerical tests show its ability to use large time steps and to achieve steady-state simulations. Mixed Finite Element (MFE) method is a robust numerical technique for solving elliptic and parabolic partial differential equations (PDEs). However, MFE can generate solutions with strong unphysical oscillations and/or large numerical diffusion for hyperbolic type PDEs. For its part, Discontinuous Galerkin (DG) finite element method is well adapted to solve hyperbolic systems and can accurately reproduce solutions involving sharp fronts. Therefore, the combination of DG and MFE is a good strategy for solving hyperbolic/parabolic problems such as advection – diffusion/dispersion equations. The classical formulation of the two methods is based on operator and time splitting allowing for separate solutions to advection with an explicit scheme and to dispersion with an implicit scheme. However, this kind of approach has the following drawbacks: (i) it lacks efficiency, as two systems with different unknowns are solved at each time step, (ii) it induces errors generated by the splitting, (iii) it can be CPU wise-expensive because of the CFL constraint, and (iv) it cannot be employed for steady-state transport simulations. To overcome these difficulties, we develop in this work a fully implicit edge/face centered DG-MFE formulation where the two methods share the same unknowns. In this formulation, the DG method is developed on lumping regions associated with the mesh edges/faces instead of mesh elements. Thus, the traces of concentration at mesh edges/faces, which are the Degrees Of Freedom (DOF) of the hybrid-MFE, are also part of the DOFs of the DG. The temporal discretization is based on the Crank–Nicolson method for both advection and dispersion. Numerical tests are performed to validate the new scheme by comparison against an analytical solution and to show its ability to handle steady-state transport simulations. The procedure is developed for 2D triangular meshes but can easily be extended to other 2D and 3D shape elements. [ABSTRACT FROM AUTHOR]

We apply a mathematical elastic shell model to describe a human mitral valve based on its geometric and mechanical properties. Specifically, we adopt an elliptic variational model called Koiter's equation to simulate the human mitral valve. Then, we provide a conforming finite element method to compute the deformation of the mitral valve. The numerical results show that the proposed mathematical model simulates well the human mitral valve. [ABSTRACT FROM AUTHOR]

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of Y. Efendiev et al. (2011) [22]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis. • We clarify and simplify the numerical analysis previously presented in [J. Comput. Phys. 230 (2011), 937-955]. • We design and analyze GMsFEM spaces constructed using local Neumann and Dirichlet eigenvalues problems. • We present an example with no optimal GMsFEM convergence rate when only Neumann eigenvalues are used. • A novel and general regularity numerical analysis tool for the high-contrast multiscale elliptic problems is presented. • We present numerical results with realistic high-contrast multiscale coefficients. [ABSTRACT FROM AUTHOR]

We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin methods, weak Galerkin methods, and hybridized versions of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the m ultigrid algorithm. [ABSTRACT FROM AUTHOR]

I. Smears and E. Süli designed and analyzed a discontinuous Galerkin finite element method for the approximation of solutions to elliptic partial differential equations in nondivergence form. The results were proven, based on the assumption that the computational domain was convex and polytopal. In this paper, we extend this framework, allowing for Lipschitz continuous domains with piecewise curved boundaries. [ABSTRACT FROM AUTHOR]

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ ℝd. The multilevel algorithm QL* which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math.15 (2015) 411–449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math.141 (2019) 63–102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput.86 (2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ ℝd, d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space. [ABSTRACT FROM AUTHOR]

Abstract In this article, we construct simple and convergent finite element methods for linear and nonlinear elliptic differential equations in non-divergence form with discontinuous coefficients. The methods are motivated by discrete Miranda–Talenti estimates, which relate the H 2 semi-norm of piecewise polynomials with the L 2 norm of its Laplacian on convex domains. We develop a stability and convergence theory of the proposed methods, and back up the theory with numerical experiments. [ABSTRACT FROM AUTHOR]

We present an investigation of the Residual-Free Bubble (RFB) finite element method for a class of multiscale nonlinear elliptic partial differential equations. After proposing a nonlinear version for the method, we address fundamental questions as existence and uniqueness of solutions. We also obtain a best approximation result and investigate possible linearizations that generate different versions for the method. All estimates are independent of eventual oscillations in the coefficients, and as far as we are aware, this is the first time that an analysis for the nonlinear RFB method is considered. [ABSTRACT FROM AUTHOR]