Computational Methods in Applied Mathematics (CMAM 2022 Conference, Part 2)

1 Introduction

Partial differential equations (PDEs) are fundamental tools in the mathematical modeling of many phenomena in science and engineering. For most of the models, especially for the most realistic ones, it is impossible to construct solutions of the underlying PDEs in “closed form”. Therefore, one has to resort computational methods to generate accurate approximations, which applied scientists then use to investigate the behavior of complex systems and real-world phenomena.

In this context, the biennial international conferences on Computational Methods in Applied Mathematics (CMAM), like the homonymous journal, are focused on various aspects of mathematical modeling and numerical analysis and aim at fostering cooperation between researchers working in the area of theoretical numerical analysis and applications to modeling, simulation, and scientific computing. The ninth edition of the conference (CMAM-9) took place from August 29 to September 2, 2022 at TU Wien (Vienna, Austria), and featured a rich scientific program consisting of overall 149 presentations, 15 plenary talks and 134 contributed talks, organized in 21 thematic minisymposia.

This CMAM special issue is the second one of overall two issues dedicated to the conference and collects twelve selected works from participants in CMAM-9 (for a summary of the first issue, we refer the interested reader to [6]). The central theme is the numerical analysis of partial differential equations (PDEs), with a particular focus on a posteriori error analysis and adaptivity [2, 4, 11, 12, 13], time-dependent problems [5, 8, 9, 10], as well as nonlinear, stochastic, and non-divergence form equations [1, 3, 7].

2 A Posteriori Error Analysis and Adaptivity

Adaptivity, in a broad sense, is the capability of a numerical scheme to allocate computational resources non-uniformly in order to invest more resources, where more accuracy is needed. Fundamental ingredients of any adaptive algorithm are a posteriori error estimates, which provide computable quantities, independent of the unknown solutions, that give local information on the accuracy of the numerical scheme. Besides the standard Solve-Estimate-Mark-Refine approach, other ideas promise new and efficient algorithms for adaptivity. This is particularly promising whenever the standard algorithms seem to have hit an obstacle.

The paper [2] derives and analyzes a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, the work employs quad-tree grids, which allow for local refinement in phase-space, and shows exemplary that adaptive methods can efficiently approximate discontinuous solutions. Hierarchical error estimators and error estimators based on local averaging are analyzed for adaptivity.

The article [12] investigates error identities that characterize distances between exact solutions of nonlinear evolutionary problems and approximations. The restrictions imposed are minimal and only on the function class of the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh-dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as modeling errors.

The article [4] discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by m-harmonic operators for m = 1 , 2 . Both the state and adjoint variables are discretized using nonconforming finite elements and error equivalence results at the continuous and discrete levels lead to a priori and a posteriori error estimates for the optimal control problem. The derived a posteriori error estimators fit into the general axiomatic framework for optimality and hence guarantee rate optimal convergence of the related adaptive scheme.

The work [11] proposes an adaptive mesh-refining algorithm based on a multi-level approach and derives a unified a posteriori error estimate for a class of nonlinear problems. The adaptive multi-level algorithms are proven to retain quadratic convergence of Newton’s method across different mesh levels, which can also be observed numerically. The developed framework facilitates the use of general theory established for linear problems associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of the theory, a pseudostress-velocity formulation of the Navier–Stokes equations as well as standard Galerkin formulations of semilinear elliptic equations are considered and reliable and efficient a posteriori error estimators for both approximations are derived.

The paper [13] presents an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation (NLS) that is used in combination with the time-splitting Fourier spectral method as the discretization for NLS. It proposes a new complex absorbing potential function based on high-order polynomials with an explicit formula for the coefficients in the potential function. This is then employed for adaptive parameter selection and leads to a more efficient imaginary potential function than what is known in the literature. Numerical examples show that the ansatz is significantly better than existing approaches.

3 Methods for Time-Dependent and Optimal Control Problems

Time-dependent problems are notoriously hard to solve as there is no straightforward way to discretize the underlying (space-time) domain. When it comes to optimal control problems, even less is known and the following works try to shed some fresh light on this area

The work [8] presents a method for the numerical approximation of distributed optimal control problems constrained by parabolic PDEs. It complements the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm. The final formulation fulfills the Babuska–Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, besides a stable discretization, it is an advantage of the least-square formulation that one also obtains an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem.

The work [9] is concerned with a space-time method for the vectorial wave equation, which is crucial for improving the existing theory of Maxwell’s equations and paves the way to computations of more complicated electromagnetic problems. The starting point of the proposed method is the derivation of a space-time variational formulation for the vectorial wave equation using different trial and test spaces (the so-called Petrov–Galerkin approach). Its unique solvability as well as the stability of a tensor product discretization (under a suitable CFL condition) are then established. The paper discusses also the derivation of the vectorial wave equation from the Maxwell equations in a space-time structure, taking into account Ohm’s law.

The work [10] studies a second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics. The numerical scheme preserves entropy structures and conserves the mass. The work establishes existence and uniqueness results for the discrete solutions, gives estimates for their large-time behavior and shows convergence of the scheme. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates.

The paper [5] proves the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties. A priori estimates imply the existence and uniqueness of the discrete approximation. With compactness estimates (relying on a Lions-like lemma) the work shows the convergence of the approximate solution to a weak solution of the problem.

4 Numerical Methods for Nonlinear, Stochastic, and Non-Divergence Form Equations

Numerical methods for nonlinear and stochastic equations or equations in non-standard form require specialized and often tailor-made discretizations in order to deliver cutting-edge performance. Designing numerical schemes and establishing their convergence is, particularly in this context, an exciting area of research to which the following works contribute.

The work [7] proposes novel computational multiscale methods for linear second-order elliptic PDEs in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction applies the localized orthogonal decomposition framework (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.

The work [3] studies the 3D stochastic Navier–Stokes equations on a torus. The main result is the proof of optimal convergence rates for the energy error with respect to convergence in probability. This means that the approximation converges at a rate of (up to) 1 in space and (up to) 1 2 in time. This result holds up to a possible blow-up time and the approach is based on discrete stopping times for the time-discrete solution.

The work [1] shows how to combine techniques of model order reduction and Gaussian Process Regression (GPR) to analyze curves that are commonly encountered in parametric eigenvalue problems. The method proposed therein is based on an offline-online decomposition method. First, in the offline phase, a basis of the reduced space is obtained by applying the proper orthogonal decomposition method on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Moreover, a GPR model is generated using four different Matérn covariance functions. In the online phase, the model is then used to predict both eigenvalues and eigenvectors at new parameters. Numerical experiments illustrate how the choice of each covariance function influences the performance of GPR. Moreover, the connection between GPR-based and spline methods is investigated and a comparison of the performance of GPR against linear and cubic spline methods is included. The main outcome of the work is that GPR seems to outperform other methods for functions with a certain regularity.