Band 25, Heft 3 - Special Issue Dedicated to the 8th Chinese–German Workshop on Computational and Applied Mathematics, Guest Editors: Malte Braack, Jun Hu, Guido Kanschat and Xuejun Xu

The Biot problem of poroelasticity is extended by Signorini contact conditions. The resulting Biot contact problem is formulated and analyzed as a two field variational inequality problem of a perturbed saddle point structure. We present an a priori error analysis for a general as well as for an hp -FE discretization including convergence and guaranteed convergence rates for the latter. In particular, these rates are always optimal in the discretization parameters for the fluid pressure, and can be linear for the lowest order h -version. Moreover, we state a residual based a posteriori error estimator. Numerical results underline our theoretical findings and show that optimal, in particular exponential, convergence rates can be achieved by adaptive schemes for two-dimensional problems.

This paper addresses the challenge of proving the existence of solutions for nonlinear equations in Banach spaces, focusing on the Navier–Stokes equations and their discretizations. Traditional methods, such as monotonicity-based approaches and fixed-point theorems, often face limitations in handling general nonlinear operators or finite element discretizations. A novel concept, mapped coercivity, provides a unifying framework to analyze nonlinear operators through a continuous mapping. We apply these ideas to saddle-point problems in Banach spaces, emphasizing both infinite-dimensional formulations and finite element discretizations. Our analysis includes stabilization techniques to restore coercivity in finite-dimensional settings, ensuring stability and existence of solutions. For linear problems, we explore the relationship between the inf-sup condition and mapped coercivity, using the Stokes equation as a case study. For nonlinear saddle-point systems, we extend the framework to mapped coercivity via surjective mappings, enabling concise proofs of existence of solutions for various stabilized Navier–Stokes finite element methods. These include Brezzi–Pitkäranta, a simple variant, and local projection stabilization (LPS) techniques, with extensions to convection-dominant flows. The proposed methodology offers a robust tool for analyzing nonlinear PDEs and their discretizations, bypassing traditional decompositions and providing a foundation for future developments in computational fluid dynamics.

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.

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.

We apply duality theory to discretized convex minimization problems to obtain computable guaranteed upper bounds for the distance of given discrete functions and the exact discrete minimizer. Furthermore, we show that the discrete duality framework extends convergence results for the Kačanov scheme to a broader class of problems.

This paper introduces a mixed finite element method for the problem of two coupled plates with mixed boundary conditions and rigid junction conditions. By introducing the union of stresses and moments as an independent variable, which is of significant interest in practical applications, a mixed formulation is developed and its well-posedness is established. To avoid complications with the direct use of trace operators, the theory of densely defined operators in Hilbert spaces is employed to determine a suitable space that incorporates boundary and junction conditions for this variable. Based on the mixed formulation, a mixed finite element method is presented along with an illustrative example. The discrete stability and the a priori analysis for the mixed finite element method are proved under some assumptions. Numerical tests demonstrate the theoretical results.

In [G. Huang, S. Boscarino and T. Xiong, High order asymptotic preserving and well-balanced schemes for the shallow water equations with source terms, Commun. Comput. Phys. 35 2024, 5, 1229–1262], we proposed a class of high-order asymptotic preserving (AP) finite difference weighted essentially non-oscillatory (WENO) schemes for solving the shallow water equations (SWEs) with bottom topography and Manning friction, utilizing a penalization technique inspired by [S. Boscarino, P. G. LeFloch and G. Russo, High-order asymptotic-preserving methods for fully nonlinear relaxation problems, SIAM J. Sci. Comput. 36 2014, 2, A377–A395]. Although the added weighted diffusive term enhanced stability, it increased computational cost and slowed down the convergence rate in the intermediate regime between convection and diffusion. In this paper, we extend our previous study by removing the penalization while preserving the AP property. To achieve this, we employ a high order semi-implicit implicit-explicit Runge–Kutta (SI-IMEX-RK) time discretization, coupled with high-order WENO reconstructions for first-order derivatives and central difference schemes for second-order spatial derivatives. This combination yields a class of fully high-order schemes. Theoretical analysis and numerical experiments demonstrate that the proposed schemes retain AP, asymptotically accurate and well-balanced properties, while offering higher computational efficiency compared to our previous scheme in Huang, Boscarino and Xiong (2024), especially in the intermediate regime between convection and diffusion. Moreover, treating the momentum in the friction terms implicitly is essential for preserving the AP property; otherwise, the scheme fails to converge to the limiting equations. This indicates that implicit treatment of Manning friction is necessary for the stability of the method.

In this paper, we observe an interesting phenomenon for a hybridizable discontinuous Galerkin (HDG) method for eigenvalue problems. Specifically, using the same finite element method, we may achieve both upper and lower eigenvalue bounds simultaneously, simply by the fine tuning of the stabilization parameter. Based on this observation, a high accuracy algorithm for computing eigenvalues is designed to yield higher convergence rate at a lower computational cost. Meanwhile, we demonstrate that certain type of HDG methods can only provide upper bounds. As a by-product, the asymptotic upper bound property of the Brezzi–Douglas–Marini mixed finite element is also established. Numerical results supporting our theory are given.

This paper presents and analyzes energy stable local discontinuous Galerkin (LDG) methods for solving the cubic nonlinear shallow water wave equations, also referred to as the Camassa–Holm–Novikov (CHN) equations. These methods are characterized by their high-order accuracy and effectiveness in addressing the intricate nonlinear terms inherent to the CHN equations. The success of the proposed numerical schemes is due to the careful selection of the numerical fluxes at the cell boundaries, which are crucial for ensuring stability and accuracy. We employ a specific interpolation function to derive optimal a priori error estimates and provide a comprehensive analysis of the quadratic Taylor expansion of the cubic nonlinearity. In addition, our numerical schemes show excellent performance for different peakon solutions. Numerical experiments validate the accuracy and efficiency of the proposed methods, demonstrating their effectiveness in practical applications.

We discuss vertex patch smoothers as overlapping domain decomposition methods for fourth order elliptic partial differential equations. We show that they are numerically very efficient and yield high convergence rates. Furthermore, we discuss low rank tensor approximations for their efficient implementation. Our experiments demonstrate that the inexact local solver yields a method which converges fast and uniformly with respect to mesh refinement and polynomial degree. The multiplicative smoother shows superior performance in terms of solution efficiency, requiring fewer iterations in both two- and three-dimensional cases. Additionally, the solver infrastructure supports a mixed-precision approach, executing the multigrid preconditioner in single precision while performing the outer iteration in double precision, thereby increasing throughput by up to 70 %.

This paper develops high-order accurate positivity-preserving (PP) and well-balanced (WB) finite volume schemes for the ten-moment Gaussian closure equations with source terms based on the prior knowledge of the equilibrium states, extending the work on the discontinuous Galerkin method in [J. Wang, H. Tang and K. Wu, High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms, J. Comput. Phys. 519 2024, Article ID 113451]. The semi-discrete schemes are constructed based on the Harten-Lax-van Leer-Contact (HLLC) flux with modified solution states, along with suitable discretization and decomposition of the source terms. The fully-discrete schemes obtained by the explicit strong-stability-preserving Rung-Kutta time discretizations (including the forward Euler) can be proved to maintain the WB property for a given known hydrostatic equilibrium state. A rigorous PP analysis for the fully-discrete schemes is provided, based on several key properties of the HLLC flux and the admissible state set as well as the geometric quasilinearlization (GQL) approach [K. Wu and C.-W. Shu, Geometric quasilinearization framework for analysis and design of bound-preserving schemes, SIAM Rev. 65 2023, 4, 1031–1073] and [K. Wu and H. Tang, Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations, Math. Models Methods Appl. Sci. 27 2017, 10, 1871–1928] transforming complex nonlinear constraints in the admissible state set into simple linear ones. Based on several newly introduced properties of the HLLC flux, we may not decompose the high-order schemes into a convex combination of the “first-order schemes”, which permits us to skip the proof of the PP property for the first-order scheme and directly analyze the high-order schemes. Consequently, the present PP analysis is more simple and direct compared to that in Wang, Tang and Wu (2024). Several numerical experiments validate the high-order accuracy, WB and PP properties as well as the high resolution of the proposed schemes.

In this paper, we design a staggered discontinuous Galerkin method for the wave propagation in poroelastic media on general polygonal meshes. The proposed method is robust with respect to the shape of the grids and can handle hanging nodes simply. The scheme shows great advantage in handling problems with complex geometries. The scheme is constructed based on the first-order hyperbolic velocity-stress system of the governing equations (i.e., Biot’s equations). Staggered continuities are imposed for the construction of the approximation spaces, as such penalty term is not needed in contrast to other DG methods. The symmetry of stress is weakly enforced via the introduction of a suitable Lagrange multiplier. The stability and convergence error estimates are analyzed. Several numerical experiments are carried out to test the performances of the proposed scheme. Numerical experiments confirm that the proposed scheme can handle polygonal elements with arbitrarily small edges without losing convergence order.