arXiv subjects

Philip L. Lederer

Publications and source records attributed to Philip L. Lederer.

At least 19 recordsLinked to original sources

arXiv2026

A mass-conserving stress-yielding formulation for the Stokes equation with exact stress symmetry

We introduce a new finite element discretization for a mixed formulation of the two-dimensional incompressible Stokes equations with symmetric viscous stresses. The method is based on a mass-conserving stress-yielding formulation (Gopalakrishnan, Lederer, Schöberl; A mass conserving mixed stress formulation for the Stokes equations; IMA J. Numer. Anal., Vol 40, 2020) with symmetric stresses, with the key novelty that the symmetry of the stress tensor is enforced exactly as an intrinsic property of the discrete space. This space is constructed on the Clough--Tocher macroelement split and consists of matrix-valued functions whose normal-tangential components across macroelement interfaces are continuous. Since the exact symmetry is enforced via local polynomial bubble functions on the subelements, which can be eliminated via static condensation, the number of globally coupled degrees of freedom coincide with those of the original mass-conserving stress-yielding formulation. The resulting pressure-robust method is stable without requiring any additional enrichment. We establish optimal convergence rates for all the variables and present numerical experiments that confirm the theoretical results.

math.NA↗

arXiv2026

Pressure-robustness by commuting interpolation operators for Stokes discretizations with continuous pressures

Common finite element discretizations for the incompressible Stokes equations with a continuous pressure approximation -- the MINI element, the Taylor--Hood element, or stabilized equal-order elements -- are not pressure-robust: the velocity error is polluted by the pressure best-approximation error, multiplied by the inverse viscosity. The established remedy replaces the test function in the momentum equation by a divergence-preserving reconstruction operator; for continuous pressure approximations this construction is cumbersome, requiring the solution of local problems (Lederer, Linke, Merdon & Schöberl, SIAM J. Numer. Anal., 2017). We propose a simpler alternative: interpolating the force by a Nédélec interpolation operator of suitable order instead of reconstructing the test function. Exploiting the commuting diagram property linking Nédélec and Lagrange interpolation operators, we show that the resulting method is pressure-robust for any $H^1$-conforming, continuous-pressure discretization. We give the complete analysis for the MINI element, and then for stabilized equal-order $P^kP^k$ elements of arbitrary degree $k$, proving that the interpolation-induced consistency error is pressure-robust and converges of optimal order. We validate the theory by several numerical examples.

math.NA↗

arXiv2026

Pressure-robustness for the axisymmetric Stokes problem by velocity reconstruction

This paper studies pressure-robustness for the axisymmetric Stokes problem. The transformation to cylindrical coordinates requires that the radially weighted velocity is divergence-free in the classical sense. Consequently, traditional divergence-free finite element methods from the Cartesian setting -- even if inf-sup stable -- are in general not divergence-free in the axisymmetric formulation. We therefore explore the approach that restores pressure-robustness via reconstruction operators for a low-order Bernardi--Raugel discretization. We show that an application of standard interpolation operators from the Cartesian setting to radially weighted test functions works in principle, but it lacks properties needed to derive optimal consistency error estimates. To address this, we introduce a reconstruction operator into a finite element space spanned by Raviart--Thomas functions that are modified such that they vanish on the rotation axis. This vanishing-on-axis property is the key to obtain optimal consistency error estimates. Numerical examples demonstrate the overall feasibility of the approach and include cases where the vanishing-on-axis property yields significantly better results.

math.NA↗

arXiv2026

IMEX Schemes for Compressible Flow using Hybridizable Discontinuous Galerkin Methods

In this work, we develop a geometry-split implicit-explicit (IMEX) framework for the compressible flow equations, wherein stiff regions are treated via an implicit hybridizable discontinuous Galerkin (HDG) method, while non-stiff regions are treated via an explicit discontinuous Galerkin (DG) method. Two implicit formulations are investigated: a mixed HDG method (HDG-MX) and a primal interior-penalty HDG method (HDG-IP). The spatial coupling between the implicit and explicit solutions is achieved in a conservative manner by appropriate interface conditions, while the temporal synchronization is maintained through the use of additive Runge-Kutta (ARK) schemes. We provide a detailed discussion on the computational performance of the resulting IMEX schemes. Verification and validation over a range of numerical experiments confirm that the proposed IMEX schemes achieve high-order accuracy in both space and time. Performance studies further indicate that the approach effectively alleviates geometry-induced stiffness and can provide speedups of up to approximately 50 relative to a fully explicit DG scheme, provided that the implicit region is chosen appropriately.

math.NA↗

arXiv2026

Embedded Trefftz DG method for steady Navier-Stokes flow. Part II: Nonlinear problem

We develop and analyze an embedded Trefftz-DG method for the steady incompressible Navier-Stokes equations, based on the reduced Oseen discretization from Part I. The main difficulty is that the reduced Trefftz space depends on the convection field, so successive Picard iterates live in different discrete spaces. We address this by constructing projections between convection-dependent Trefftz spaces and using them to control the reduced Oseen solution map. Under suitable resolution and small-data assumptions, we prove existence of discrete solutions, uniqueness, and convergence of the Picard iteration. We also derive an a priori error analysis by relating the method to the underlying DG discretization, thereby inheriting convergence properties from compatible DG Navier-Stokes analyses. Numerical experiments on standard incompressible-flow benchmarks illustrate the theory.

math.NA↗

arXiv2026

Embedded Trefftz DG method for steady Navier-Stokes flow. Part I: Oseen linearization

We develop an embedded Trefftz-DG method for the Oseen problem and prove a complete stability and quasi-optimality theory in standard DG norms. The key ingredient is a construction of a suitable local complement space to the Trefftz space, on which the Oseen operator is stably invertible. We also derive a reduced formulation of the method, the resulting system is posed in terms of the velocity unknown only, a crucial step in the analysis especially for the nonlinear Navier-Stokes problem in Part II.

math.NA↗

arXiv2025

A bound-preserving and conservative enriched Galerkin method for elliptic problems

We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial over-penalization of the discrete solution's jumps to obtain optimal convergence. To avoid the ill-conditioning issues that arise in over-penalized schemes, we introduce an involved splitting approach that separates the system of equations for the discontinuous solution part from the system of equations for the continuous solution part, yielding well-behaved subproblems. We prove the existence of discrete solutions and optimal error estimates, which are validated numerically.

math.NA↗

arXiv2025

Embedded Trefftz DG framework for the analysis of discretizations with local-global decompositions

This paper presents a framework for the analysis of discretization methods based on the decomposition into local and global problems. We apply the framework to provide a comprehensive error analysis for the embedded Trefftz discontinuous Galerkin method, for a wide range of second-order scalar elliptic partial differential equations and a scalar reaction-advection problem. We also analyze quasi-Trefftz methods with our framework, presenting the first optimal error bounds in weaker norms.

math.NA↗

arXiv2025

On positivity preservation of hybrid discontinuous Galerkin methods on hypergraphs

Hybrid finite element methods, particularly hybridized discontinuous Galerkin (HDG) methods, are efficient numerical schemes for discretizing the diffusion equation, which encompasses two main physical principles: mass conservation and positivity preservation. While the former has been extensively analyzed in the literature, this paper investigates the latter. We state a theorem that guarantees the positivity of both the bulk and skeleton approximations to the primary unknown (concentration) and provide counterexamples for nonpositive discretizations. The theoretical findings are confirmed by numerical experiments.

math.NA↗

arXiv2025

Characteristic boundary conditions for Hybridizable Discontinuous Galerkin methods

In this work we introduce the concept of characteristic boundary conditions (CBCs) within the framework of Hybridizable Discontinuous Galerkin (HDG) methods, including both the Navier-Stokes characteristic boundary conditions (NSCBCs) and a novel approach to generalized characteristic relaxation boundary conditions (GRCBCs). CBCs are based on the characteristic decomposition of the compressible Euler equations and are designed to prevent the reflection of waves at the domain boundaries. We show the effectiveness of the proposed method for weakly compressible flows through a series of numerical experiments by comparing the results with common boundary conditions in the HDG setting and reference solutions available in the literature. In particular, HDG with CBCs show superior performance minimizing the reflection of vortices at artificial boundaries, for both inviscid and viscous flows.

math.NA↗

arXiv2025

Evaporating sessile droplets: solutal Marangoni effects overwhelm thermal Marangoni flow

When an evaporating water droplet is deposited on a thermally conductive substrate, the minimum temperature will be at the apex due to evaporative cooling. Consequently, density and surface tension gradients emerge within the droplet and at the droplet-gas interface, giving rise to competing flows from, respectively, the apex towards the contact line (thermal-buoyancy-driven flow) and the other way around (thermal Marangoni flow). In small droplets with a diameter below the capillary length, the thermal Marangoni effects are expected to dominate over thermal buoyancy ("thermal Rayleigh") effects. However, contrary to these theoretical predictions, our experiments mostly show a dominant circulation from the apex towards the contact line, indicating a prevailing of thermal Rayleigh convection. Furthermore, our experiments often show an unexpected asymmetric flow that persisted for several minutes. We hypothesise that a tiny amount of contaminants, commonly encountered in experiments with water/air interfaces, act as surfactants and counteract the thermal surface tension gradients at the interface and thereby promote the dominance of Rayleigh convection. Our finite element numerical simulations demonstrate that, under our specified experimental conditions, a mere 0.5% reduction in the static surface tension caused by surfactants leads to a reversal in the flow direction, compared to the theoretical prediction without contaminants. Additionally, we investigate the linear stability of the axisymmetric solutions, revealing that the presence of surfactants also affects the axial symmetry of the flow.

physics.flu-dyn↗

arXiv2024

High-order projection-based upwind method for simulation of transitional turbulent flows

We present a scalable, high-order implicit large-eddy simulation (ILES) approach for incompressible transitional flows. This method employs the mass-conserving mixed stress (MCS) method for discretizing the Navier-Stokes equations. The MCS method's low dissipation characteristics, combined with the introduced operator-splitting solution technique, result in a high-order solver optimized for efficient and parallel computation of under-resolved turbulent flows. We further enhance the inherent capabilities of the ILES model by incorporating high-order upwind fluxes and are examining its approximation behaviour in transitional aerodynamic flow problems. In this study, we use flows over the Eppler 387 airfoil at Reynolds numbers up to $3 \cdot 10^5$ as benchmarks for our simulations.

cs.CE↗

arXiv2024

Trefftz Discontinuous Galerkin discretization for the Stokes problem

We introduce a new discretization based on the Trefftz-DG method for solving the Stokes equations. Discrete solutions of a corresponding method fulfill the Stokes equation pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher order polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows to easily incorporate inhomogeneous right hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to standard discontinuous Galerkin Stokes discretizations and present numerical examples.

math.NA↗

arXiv2023

A discontinuous Galerkin approach for atmospheric flows with implicit condensation

We present a discontinuous Galerkin method for moist atmospheric dynamics, with and without warm rain. By considering a combined density for water vapour and cloud water, we avoid the need to model and compute a source term for condensation. We recover the vapour and cloud densities by solving a pointwise non-linear problem each time step. Consequently, we enforce the requirement for the water vapour not to be supersaturated implicitly. Together with an explicit time-stepping scheme, the method is highly parallelisable and can utilise high-performance computing hardware. Furthermore, the discretisation works on structured and unstructured meshes in two and three spatial dimensions. We illustrate the performance of our approach using several test cases in two and three spatial dimensions. In the case of a smooth, exact solution, we illustrate the optimal higher-order convergence rates of the method.

math.NA↗

arXiv2023

Gradient-robust hybrid DG discretizations for the compressible Stokes equations

This paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an H(div)-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier-Stokes equations.

math.NA↗

arXiv2023

Divergence-conforming velocity and vorticity approximations for incompressible fluids obtained with minimal facet coupling

We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (HDG) method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space for approximating the velocity and the lowest order Raviart-Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

math.NA↗

arXiv2023

Analysis of Weakly Symmetric Mixed Finite Elements for Elasticity

We consider mixed finite element methods for linear elasticity where the symmetry of the stress tensor is weakly enforced. Both an a priori and a posteriori error analysis are given for several known families of methods that are uniformly valid in the incompressible limit. A posteriori estimates are derived for both the compressible and incompressible cases. The results are verified by numerical examples.

math.NA↗

arXiv2023

High-order projection-based upwind method for implicit large eddy simulation

We assess the ability of three different approaches based on high-order discontinuous Galerkin methods to simulate under-resolved turbulent flows. The capabilities of the mass conserving mixed stress method as structure resolving large eddy simulation solver are examined. A comparison of a variational multiscale model to no-model or an implicit model approach is presented via numerical results. In addition, we present a novel approach for turbulent modeling in wall-bounded flows. This new technique provides a more accurate representation of the actual subgrid scales in the near wall region and gives promising results for highly under-resolved flow problems. In this paper, the turbulent channel flow and periodic hill flow problem are considered as benchmarks for our simulations.

physics.flu-dyn↗