Abstract We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art semidefinite programming solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.

Abstract This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. It is also shown how to approximate a Hessian-vector product with a minimal number of function evaluations. These approximations are obtained using the techniques called generalized simplex Hessian and generalized centred simplex Hessian. We show how to choose the matrices of directions involved in the computation of these two techniques, depending on the entries of the Hessian of interest. We discuss the number of function evaluations required in each case and develop a general formula to approximate all order-$P$ partial derivatives. Since only function evaluations are required to compute the methods discussed in this paper they are suitable for use in derivative-free optimization methods.

Abstract We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $\varOmega \subset{\mathbb{R}}^{2}$ with a finite number of straight edges. In particular, we analyse the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $\varOmega $, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement towards the corners of $\varOmega $. For a sequence of discrete solutions generated by the ILG solver with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution we prove exponential convergence in $\text{H}^{1}(\varOmega )$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.

Abstract We prove rigorous a-posteriori error estimates for first-order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by Bressan, Chiri and Shen, a new way to localize residuals and a novel method to compute negative-order norms of these local residuals. Computing negative-order norms becomes possible by suitably projecting test functions onto a finite dimensional space. Numerical experiments show that the error estimator converges with the rate predicted by a-priori error estimates.

Abstract We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $\mathcal{O}(\varepsilon)$ around the interface; this allows the weak forms to be integrated uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretization of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We, in turn, prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis.

Abstract A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions by the example of the Poisson model problem with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables the penalization of the jump with the natural power of the mesh size in the least-squares functional. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments illustrate the importance of the proposed weighting and exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees.

Abstract One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by the under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. We propose a new way to update the quadratic model by minimizing the $H^{2}$ norm of the difference between neighboring quadratic models. The motivation for applying the $H^{2}$ norm is given. The theoretical properties of our new updating technique are also presented. We propose the projection in the sense of $H^{2}$ norm and the interpolation error analysis of our model function. We obtain the coefficients of the quadratic model function using the Karush–Kuhn–Tucker (KKT) conditions. Numerical results show the advantages of our model on the test set considered, and the derivative-free algorithms based on our least $H^{2}$ norm updating quadratic model functions can solve test problems with fewer function evaluations than the algorithm based on the least Frobenius norm updating model and the other compared methods.

Abstract In this paper, we develop a fully nonconforming (both $H(\operatorname{grad}\operatorname{curl})$-nonconforming and $H(\operatorname{curl})$-nonconforming) finite element on cubical meshes that can fit into the Stokes complex. The newly proposed element has 24 degrees of freedom. Unlike the fully $H(\operatorname{grad}\operatorname{curl})$-nonconforming tetrahedral finite elements in Huang (2023, Nonconforming finite element stokes complexes in three dimensions. Sci. China Math., 66, 1879–1902), the element in this paper yields a robust finite element method to solve the singularly perturbed quad-curl problem. To confirm this, we prove the optimal convergence of order $\mathscr O(h)$ for a fixed parameter $\varepsilon $ and the uniform convergence of order $\mathscr O(h^{1/2})$ for any value of $\varepsilon $ in the sense of an $H(\operatorname{grad}\operatorname{curl})$-norm. When $\varepsilon $ tends to 0, we employ the Nitsche’s method to improve the convergence order from $\mathscr O(h^{1/2})$ to $\mathscr O(h)$. Some numerical examples are used to verify the validity of the theoretical analysis.

Abstract In highly diffusion regimes when the mean free path $\varepsilon $ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon ^{-1}$ contribution that leads to a nonuniform convergence for small $\varepsilon $. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a priori estimates for the scaled spherical harmonic ($P_{N}$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some additional assumptions, its solutions converge uniformly in $\varepsilon $ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left (1+\mathcal{O}(\varepsilon )\right )h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.

Abstract The dynamics of the magnetization in ferromagnetic materials is governed by the Landau–Lifshitz–Gilbert equation, which is highly nonlinear with the nonconvex sphere constraint $|{\textbf{m}}|=1$. A crucial issue in designing numerical schemes is to preserve this sphere constraint in the discrete level. A popular numerical method is the normalized tangent plane finite element method (NTP-FEM), which was first proposed by Alouges and Jaisson and later, applied for solving various practical problems. Since the classical energy approach fails to be applied directly to the analysis of this method, previous studies only focused on the convergence and until now, no any error estimate was established for such an NTP-FEM. This paper presents a rigorous error analysis and establishes the optimal $H^{1}$ error estimate. Numerical results are provided to confirm our theoretical analysis.