Abstract Large linear systems are ubiquitous in modern computational science and engineering. Their efficient solution is frequently challenging, especially in conjunction with problems described by parametric PDEs, where many such systems have to be solved. The primary approach for solving large linear systems is the use of Krylov subspace iterative methods with well-designed preconditioners. Recently, graph neural networks (GNNs) have been used in conjunction with solving parametric PDEs, and have been shown to be promising tool for designing preconditioners. Nevertheless, the GNN-based preconditioners reported in the literature have worse effect on system’s spectrum than analogous preconditioners from classical linear algebra. Here we employ well-established preconditioners from linear algebra as starting point for training GNN to obtain preconditioners that yield a more substantial reduction in the condition number of the systems when compared to classical preconditioners. Numerical experiments demonstrate the efficiency of our approach in comparison to classical and known neural network-based methods for parametric PDEs. In addition, a heuristic justification for the loss function employed in this study is provided, and we demonstrate that preconditioners obtained by learning with this loss function reduce the condition number in a way that is more desirable for the conjugate gradient iterative method.

Abstract The paper is concerned with a posteriori estimates for approximations of boundary value problems generated by the spectral fractional Laplace operator. The derivation is based upon the Stinga–Torrea extension, which generalizes the Caffarelli–Silvestre extension and transfers the corresponding nonlocal problem in a bounded domain to a local problem of higher dimensionality. A posteriori estimates are first derived for this local problem. Two–sided error bounds for the original problem follow from them. The estimates are fully computable and contain no conditions and constants depending on a method or mesh used to compute an approximation. They are valid for any energy admissible approximation of the extended problem.

Abstract In this study, we propose a parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions. This model describes the energy-driven motion of a surface cluster whose distributional solution was studied by Garcke and Sturzenhecker. We approximate the weak formulation of this sharp interface model by an unfitted finite element method that uses parametric elements for the representation of the moving interfaces. We establish existence and uniqueness of the discrete solution and prove unconditional stability of the proposed scheme. Moreover, a modification of the original scheme leads to a structure-preserving variant, in that it conserves the discrete analogue of a quantity that is preserved by the classical solution. Some numerical results demonstrate the applicability of our introduced schemes.

Abstract In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate for the L 2 {L^{2}} -norm of the space-time discretization of SHE by a semi-implicit Euler scheme with respect to time and a TPFA scheme with respect to space and the variational solution of SHE. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.

Abstract We present the Super-Localized Orthogonal Decomposition (SLOD) method for the numerical homogenization of linear elasticity problems with multiscale microstructures modeled by a heterogeneous coefficient field without any periodicity or scale separation assumptions. Compared to the established Localized Orthogonal Decomposition (LOD) and its linear localization approach, SLOD achieves significantly improved sparsity properties through a nonlinear superlocalization technique, leading to computationally efficient solutions with significantly less oversampling – without compromising accuracy. We generalize the method to vector-valued problems and provide a supporting numerical analysis. We also present a scalable implementation of SLOD using the deal.II finite element library, demonstrating its feasibility for high-performance simulations. Initial numerical experiments indicate the potential of SLOD for addressing computational challenges in multiscale elasticity.

Abstract The solution operator in a Petrov Galerkin scheme in Hilbert spaces is an oblique projection with quasi-best approximation property. The latter estimate involves a multiplicative constant and the best-possible of those is the target of the note: We present a new direct proof of the formula of the quasi-best approximation constant and avoid the direct application of the Kato lemma. In fact, our characterisation leads to another proof of the Kato oblique projection lemma. The abstract result in Hilbert spaces is embedded in the setting of the best-approximation of conforming Petrov Galerkin schemes with a rich history that eventually led to the Tantardini–Veeser formula. A final application discusses the classical nonconforming schemes with a smoother in this framework.

Abstract For a dual-dual formulation of a frictional contact problem in linear elasticity we present a mixed finite element method based on PEERS elements and we analyze the performance of a nested Uzawa algorithm.

Abstract A potential-robust WG (weak Galerkin) method is introduced for the Maxwell equations. We obtain potential/magnetic-permeability independent error estimates. Optimal-order convergence rates are proved in both the energy norm and the L 2 {L^{2}} norm. Numerical examples verify the theory.

Abstract The free longitudinal vibrations of a rod are described by a differential equation of the form ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0} , where P ⁢ ( x ) {P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀 ⁢ Y = Λ ⁢ 𝐁 ⁢ Y {\mathbf{A}Y=\Lambda\mathbf{B}Y} , where 𝐀 {\mathbf{A}} and 𝐁 {\mathbf{B}} are Jacobi and diagonal matrices dependent to cross section P ⁢ ( x ) {P(x)} , respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section P ⁢ ( x ) {P(x)} by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is O ⁢ ( h 2 ) {O(h^{2})} .