Václav Kučera

Václav Kučera, Chi-Wang Shu

In this paper we derive a priori $L^\infty(L^2)$ and $L^2(L^2)$ error estimates for a linear advection-reaction equation with inlet and outlet boundary conditions. The goal is to derive error estimates for the discontinuous Galerkin (DG) method that do not blow up exponentially with respect to time, unlike the usual case when Gronwall's inequality is used. While this is possible in special cases, such as divergence-free advection fields, we take a more general approach using exponential scaling of the exact and discrete solutions. Here we use a special scaling function, which corresponds to time taken along individual pathlines of the flow. For advection fields, where the time massless particles carried by the flow spend inside the spatial domain is uniformly bounded from above by some $\widehat{T}$, we derive $O(h^{p+1/2})$ error estimates where the constant factor depends only on $\widehat{T}$, but not on the final time $T$. This can be interpreted as applying Gronwall's inequality in the error analysis along individual pathlines (Lagrangian setting), instead of physical time (Eulerian setting).

Jiří Szotkowski, Václav Kučera, Chi-Wang Shu et al.

We establish a simple, rigorous, and easy to implement connection between the classical continuous finite element method (FEM) and the discontinuous Galerkin (DG) method for Poisson's problem. The key idea is to insert a vanishing-thickness layer of "dummy" elements along cell interfaces. By modifying the diffusion coefficient on these elements to be proportional to their thickness, we prove the FEM formulation converges to Babuška-Zlámal DG with trapezoidal edge quadrature. The scheme is trivial to implement by (i) a mesh edit that introduces degenerate interface elements and (ii) a single Jacobian threshold in an otherwise unmodified FEM code to handle the degenerate elements via the tempered finite element (TFEM) framework. We provide a rigorous derivation of the resulting TFEM-DG scheme, prove optimal $H^1$ and $L^2$ error estimates, and present numerical experiments in 2D and 3D. The method allows for simple implementation of DG in a FEM code and even adaptive element-by-element switching between FEM and DG with minimal coding effort. The framework is readily extensible, as we will demonstrate in a companion paper dedicated to evolutionary nonlinear first-order hyperbolic systems.

Václav Kučera

In this paper we derive a necessary condition for finite element method (FEM) convergence in $H^1(Ω)$ as well as generalize known sufficient conditions. We deal with the piecewise linear conforming FEM on triangular meshes for Poisson's problem in 2D. In the first part, we prove a necessary condition on the mesh geometry for $O(h^α)$ convergence in the $H^1(Ω)$-seminorm with $α\in[0,1]$. We prove that certain structures, bands consisting of neighboring degenerating elements forming an alternating pattern cannot be too long with respect to $h$. This is a generalization of the Babu\v ska-Aziz counterexample and represents the first nontrivial necessary condition for finite element convergence. Using this condition we construct several counterexamples to various convergence rates in the FEM. In the second part, we generalize the maximum angle and circumradius conditions for $O(h^α)$ convergence. We prove that the triangulations can contain many elements violating these conditions as long as their maximum angle vertexes are sufficiently far from other degenerating elements or they form clusters of sufficiently small size. While a necessary and sufficient condition for $O(h^α)$ convergence in $H^1(Ω)$ remains unknown, the gap between the derived conditions is small in special cases.

Matěj Gajdoš, Ondřej Brichta, Václav Kučera

We study how inexact nonlinear solvers lead to a loss of exact symplecticity in the Symplectic Euler (SE) and Stormer-Verlet (SV) schemes when applied to general nonseparable Hamiltonian systems. These schemes are implicit and require nonlinear solvers in practice. Here, we consider a fixed number $M$ of fixed-point iterations (FPI). While SE is exactly symplectic under exact solves, a finite $M$ gives only pseudo-symplecticity. Compared to previous results, we provide a more qualitative, block-wise characterization of the induced pseudo-symplecticity by analyzing the resulting perturbations to the matrix of symplectic structure $J$. We prove that the perturbed matrix $\tilde{J}$ is skew-symmetric, that one diagonal block vanishes identically (depending on the SE variant), and that the remaining blocks are $O(h^{M+1})$ perturbations of their counterparts in $J$, with time step $h$. A quadratic Hamiltonian example shows these bounds are sharp. Extending to compositions, we quantify how SV inherits distinct decay orders across different blocks of the symplectic defect. As a corollary, we show that the perturbation of volume preservation in phase space arises solely from the off-diagonal blocks of $\tilde{J}$, and we bound the induced energy error along trajectories. Numerical experiments on a tokamak magnetic-field Hamiltonian, where q-implicit SE is fully nonlinear (requiring FPI) but p-implicit SE is linearly implicit, confirm the sharpness of the theory and highlight the gap to the exactly symplectic counterpart.