Trefftz Methods in NGSolve

Trefftz Workshop 7-9 September 2026 · Vienna, Austria¶

https://trefftz2026.univie.ac.at/

Overview¶

This book is a demo for Trefftz and embedded Trefftz methods in NGSolve/NGSTrefftz. The package provides implementations of

Trefftz spaces and quasi-Trefftz spaces for several PDEs
Trefftz-DG on tent-pitched meshes for the acoustic wave equation using ngstents
Embedded Trefftz method
Constraint-defined finite element spaces as in ngstSpaceKit

Follow along: https://github.com/PaulSt/ngst26

We focus the examples on results from the recent publications:

Embedded Trefftz DG method for steady Navier-Stokes flow. Part II: Nonlinear problem
Paul Stocker, Igor Voulis, Christoph Lehrenfeld, Philip L. Lederer
Embedded Trefftz DG method for steady Navier-Stokes flow. Part I: Oseen linearization
Paul Stocker, Igor Voulis, Christoph Lehrenfeld, Philip L. Lederer
Embedded Trefftz DG method for reaction-diffusion problems on anisotropic meshes
Sergio Gómez, Chiara Perinati, Paul Stocker, Igor Voulis
Embedded Trefftz DG method for the Helmholtz equation
Paul Stocker, Igor Voulis
On the Conforming Trefftz Finite Element Method and Applications
Johann Carl Meyer, Master’s thesis

Contributors¶

from datetime import datetime, timedelta
from itertools import accumulate

import matplotlib.dates as mdates
import matplotlib.pyplot as plt

commit_csv = """month,count
2017-08,6
2017-09,28
2017-10,19
2017-11,15
2017-12,13
2018-01,9
2018-02,4
2018-03,11
2018-04,14
2018-05,1
2018-06,19
2018-07,24
2018-08,16
2018-09,11
2018-10,54
2018-11,17
2018-12,22
2019-01,41
2019-02,11
2019-03,4
2019-04,42
2019-05,8
2019-06,2
2019-07,0
2019-08,2
2019-09,0
2019-10,8
2019-11,15
2019-12,5
2020-01,31
2020-02,13
2020-03,1
2020-04,3
2020-05,6
2020-06,28
2020-07,5
2020-08,8
2020-09,14
2020-10,3
2020-11,14
2020-12,6
2021-01,4
2021-02,1
2021-03,5
2021-04,0
2021-05,0
2021-06,11
2021-07,0
2021-08,34
2021-09,4
2021-10,0
2021-11,34
2021-12,23
2022-01,32
2022-02,34
2022-03,23
2022-04,8
2022-05,2
2022-06,6
2022-07,0
2022-08,14
2022-09,6
2022-10,27
2022-11,5
2022-12,0
2023-01,4
2023-02,8
2023-03,3
2023-04,10
2023-05,1
2023-06,4
2023-07,15
2023-08,4
2023-09,2
2023-10,6
2023-11,10
2023-12,10
2024-01,7
2024-02,22
2024-03,13
2024-04,6
2024-05,36
2024-06,54
2024-07,36
2024-08,7
2024-09,26
2024-10,17
2024-11,6
2024-12,0
2025-01,26
2025-02,13
2025-03,12
2025-04,37
2025-05,4
2025-06,18
2025-07,3
2025-08,3
2025-09,19
2025-10,9
2025-11,6
2025-12,0
2026-01,5
2026-02,9
2026-03,5
2026-04,7
2026-05,0
2026-06,7
"""

rows = [line.split(",") for line in commit_csv.strip().splitlines()[1:]]
months = [datetime.strptime(month, "%Y-%m") for month, _ in rows]
monthly_commits = [int(count) for _, count in rows]
cumulative_commits = list(accumulate(monthly_commits))

fig, ax = plt.subplots(figsize=(10, 3.2))
ax.bar(
    months,
    monthly_commits,
    width=25,
    color="#4e79a7",
    alpha=0.85,
    label="Monthly commits",
)
ax.set_ylabel("Monthly commits")
ax.set_title("NGSTrefftz commits over time")
ax.grid(axis="y", alpha=0.25)
ax.xaxis.set_major_locator(mdates.YearLocator())
ax.xaxis.set_major_formatter(mdates.DateFormatter("%Y"))
ax.set_xlim(months[0] - timedelta(days=25), months[-1] + timedelta(days=25))

ax2 = ax.twinx()
ax2.plot(
    months,
    cumulative_commits,
    color="#e15759",
    linewidth=2.2,
    label="Cumulative commits",
)
ax2.set_ylabel("Cumulative commits")
ax2.set_ylim(0, max(cumulative_commits) * 1.08)

handles, labels = ax.get_legend_handles_labels()
handles2, labels2 = ax2.get_legend_handles_labels()
ax.legend(handles + handles2, labels + labels2, loc="upper left", frameon=False)
fig.tight_layout()