We study an embedded Trefftz discontinuous Galerkin method for the Helmholtz equation. The method starts from a polynomial DG space and enforces the Trefftz property through local constraints, avoiding an explicit construction of Trefftz basis functions. For the global coupling we use a simple symmetric interior penalty DG bilinear form. Since the resulting formulation is not coercive, stability is proved by a T-coercivity argument combined with a Schatz-type duality technique. This yields wavenumber-explicit stability, quasi-optimality, and convergence estimates in standard DG norms under an explicit mesh resolution condition.

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.

We present and analyze an embedded Trefftz discontinuous Galerkin method for reaction-diffusion problems on anisotropic meshes. The method is constructed by imposing a relaxed local Trefftz condition via an embedding into a tensor-product DG space, yielding a reduced global system while preserving the approximation properties of the underlying high-order discretization. We prove stability and quasi-optimality on anisotropic, possibly curved, quadrilateral elements, and derive anisotropic a priori error estimates. Numerical experiments for h- and hp-refinement, including curved-domain examples, validate the theoretical results.

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.

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the computational domain. Well-posedness of the discrete problem is established via coercivity in an energy norm, achieved through the Morawetz multiplier technique. We derive hp-a priori error estimates in the energy norm, which we use to prove convergence rates for standard and quasi-Trefftz polynomial spaces. Numerical experiments validate the theoretical results.

We compare additive and multiplicative domain decomposition preconditioners, without coarse correction, for three non-conforming polynomial discretizations of Helmholtz problems: discontinuous Galerkin, embedded Trefftz discontinuous Galerkin, and hybrid discontinuous Galerkin methods. The preconditioners are studied within stationary and Krylov iterative solvers. All three discretizations lead to complex-symmetric systems and local subproblems with inherited impedance-type boundary conditions. These properties enable the use of conjugate gradients and allow the local matrices to be obtained directly by restriction of the global matrix. Numerical experiments in two and three dimensions demonstrate promising performance for large-scale Helmholtz problems.

We propose and analyze a space–time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space–time meshes. Existence and uniqueness of a discrete solution are shown by means of an inf-sup condition, whose proof does not rely on polynomial inverse estimates. Moreover, for piecewise polynomial spaces satisfying an additional mild condition, we show a second inf-sup condition that provides additional control over the time derivative of the discrete solution. We derive hp-a priori error bounds based on these inf-sup conditions, which we use to prove convergence rates for standard, tensor-product, and quasi-Trefftz polynomial spaces. Numerical experiments validate our theoretical results.

We consider the approximation of weakly T-coercive operators. The main property to ensure the convergence thereof is the regularity of the approximation (in the vocabulary of discrete approximation schemes). In a previous work the existence of discrete operators T_n which converge to T in a discrete norm was shown to be sufficient to obtain regularity. Although this framework proved usefull for many applications for some instances the former assumption is too strong. Thus in the present article we report a weaker criterium for which the discrete operators T_n only have to converge point-wise, but in addition a weak T-coercivity condition has to be satisfied on the discrete level. We apply the new framework to prove the convergence of certain H^1-conforming finite element discretizations of the damped time-harmonic Galbrun's equation, which is used to model the oscillations of stars. A main ingredient in the latter analysis is the uniformly stable invertibility of the divergence operator on certain spaces, which is related to the topic of divergence free elements for the Stokes equation.

Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise 'approximate solutions' of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion-advection-reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.

In this work we compare crucial parameters for efficiency of different finite element methods for solving partial differential equations  on polytopal meshes. We consider the virtual element method (VEM) and different discontinuous Galerkin (DG) methods, namely, the Hybrid DG and Trefftz DG methods. The VEM is a conforming method, that can be seen as a generalization of the classic finite element method to arbitrary polytopal meshes. DG methods are non-conforming methods that offer high flexibility, but also come with high computational costs. Hybridization reduces these costs by introducing additional facet variables, onto which the computational costs can be transferred to. Trefftz DG methods achieve a similar reduction in complexity by selecting a special and smaller set of basis functions on each element. The association of computational costs to different geometrical entities (elements or facets) leads to differences in the performance of these methods on different grid types. This paper aims to compare the dependency of these approaches across different grid configurations.

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.

We introduce a new discretization based on a polynomial Trefftz-DG method for solving the Stokes equations. Discrete solutions of this method fulfill the Stokes equations 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 polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows us 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 other (hybrid) discontinuous Galerkin Stokes discretizations and present numerical examples.

We study the approximation properties of complex-valued polynomial Trefftz spaces for the $(d+1)$-dimensional linear time-dependent Schrödinger equation. More precisely, we prove that for the space-time Trefftz discontinuous Galerkin variational formulation proposed by Gómez, Moiola (SIAM. J. Num. Anal. 60(2): 688-714, 2022), the same $h$-convergence rates as for polynomials of degree $p$ in $(d + 1)$ variables can be obtained in a mesh-dependent norm by using a space of Trefftz polynomials of anisotropic degree. For such a space, the dimension is equal to that of the space of polynomials of degree $2p$ in $d$ variables, and bases are easily constructed.

Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise-constant. We introduce a “quasi-Trefftz” discontinuous Galerkin (DG) method for the discretisation of the acoustic wave equation with piecewise-smooth material parameters: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.

We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.

In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz-type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non-constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.

Driven by the challenging task of finding robust discretization methods for Galbrun's equation, we investigate conditions for stability and different aspects of robustness for different finite element schemes on a simplified version of the equations. The considered PDE is a second order indefinite vector-PDE which remains if only the highest order terms of Galbrun's equation are taken into account. A key property for stability is a Helmholtz-type decomposition which results in a strong connection between stable discretizations for Galbrun's equation and Stokes and nearly incompressible linear elasticity problems.

NGSTrefftz is an add-on to Netgen/NGSolve, a finite element software for the numerical treatment of partial differential equations (PDEs).The package implements Trefftz based discontinuous Galerkin (DG) methods in NGSolve.Trefftz methods reduce the number of unknowns in the discretization of PDE problems by injecting knowledge of the PDE into the approximation functions.Like NGSolve, NGSTrefftz is written in C++ and integrates seamlessly with the easy-to-use Python interface of NGSolve.Trefftz methods originate from Trefftz (1926) and have since been developed for a wide range of problems, for an overview see Zienkiewicz (1997), Hiptmair et al. (2016), Qin (2005), Li et al. (2008), Kita & Kamiya (1995).The central principle of Trefftz methods is the construction of a discrete basis of solutions to the differential operator under consideration, making the space of Trefftz functions problem dependent.In combination with finite elements the Trefftz basis is constructed locally, on each mesh element, continuity and boundary conditions are then enforced in the variational formulation.

Cross-diffusion systems are systems of nonlinear parabolic partial differential equations that are used to describe dynamical processes in several application, including chemical concentrations and cell biology. We present a space-time approach to the proof of existence of bounded weak solutions of cross-diffusion systems, making use of the system entropy to examine long-term behavior and to show that the solution is nonnegative, even when a maximum principle is not available. This approach naturally gives rise to a novel space-time Galerkin method for the numerical approximation of cross-diffusion systems that conserves their entropy structure. We prove existence and convergence of the discrete solutions, and present numerical results for the porous medium, the Fisher-KPP, and the Maxwell-Stefan problem.

Driven by the challenging task of finding robust discretization methods for Galbrun's equation, we investigate conditions for stability and different aspects of robustness for different finite element schemes on a simplified version of the equations. The considered PDE is a second order indefinite vector-PDE which remains if only the highest order terms of Galbrun's equation are taken into account. A key property for stability is a Helmholtz-type decomposition which results in a strong connection between stable discretizations for Galbrun's equation and Stokes and nearly incompressible linear elasticity problems.