ON CONVERGENCE OF THE C0 INTERIOR PENALTY METHOD FOR mTH LAPLACE EQUATION

Xianhao Zeng

doi:10.3934/cac.2025021

ON CONVERGENCE OF THE C⁰ INTERIOR PENALTY METHOD FOR mTH LAPLACE EQUATION

Xianhao Zeng^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

We study the C⁰ interior penalty method for the mth Laplace equation on a bounded Lipschitz polyhedral domain in R^d, showing, without extra regularity assumption, that the numerical solution converges to the exact solution both under discrete H^l-norms where 2 ≤ l ≤ m−1 and in H¹₀(Ω). The strategy we applied consists of three steps: carrying a compactness argument to enumerate all limit points of u_h, proving each of them identical to the unique solution to the PDE, and deducing the convergence. © 2025, American Institute of Mathematical Sciences. All rights reserved.

Original language	English
Pages (from-to)	94-108
Number of pages	15
Journal	Communications on Analysis and Computation
Volume	6
DOIs	https://doi.org/10.3934/cac.2025021
Publication status	Published - Dec 2025

Research Keywords

C0 IP method
compactness argument
convergence analysis
discontinuous Galerkin method
mth Laplace equation

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title = "ON CONVERGENCE OF THE C0 INTERIOR PENALTY METHOD FOR mTH LAPLACE EQUATION",

abstract = "We study the C0 interior penalty method for the mth Laplace equation on a bounded Lipschitz polyhedral domain in Rd, showing, without extra regularity assumption, that the numerical solution converges to the exact solution both under discrete Hl-norms where 2 ≤ l ≤ m−1 and in H10(Ω). The strategy we applied consists of three steps: carrying a compactness argument to enumerate all limit points of uh, proving each of them identical to the unique solution to the PDE, and deducing the convergence. {\textcopyright} 2025, American Institute of Mathematical Sciences. All rights reserved.",

keywords = "C0 IP method, compactness argument, convergence analysis, discontinuous Galerkin method, mth Laplace equation",

author = "Xianhao Zeng",

year = "2025",

month = dec,

doi = "10.3934/cac.2025021",

language = "English",

volume = "6",

pages = "94--108",

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TY - JOUR

T1 - ON CONVERGENCE OF THE C0 INTERIOR PENALTY METHOD FOR mTH LAPLACE EQUATION

AU - Zeng, Xianhao

PY - 2025/12

Y1 - 2025/12

N2 - We study the C0 interior penalty method for the mth Laplace equation on a bounded Lipschitz polyhedral domain in Rd, showing, without extra regularity assumption, that the numerical solution converges to the exact solution both under discrete Hl-norms where 2 ≤ l ≤ m−1 and in H10(Ω). The strategy we applied consists of three steps: carrying a compactness argument to enumerate all limit points of uh, proving each of them identical to the unique solution to the PDE, and deducing the convergence. © 2025, American Institute of Mathematical Sciences. All rights reserved.

AB - We study the C0 interior penalty method for the mth Laplace equation on a bounded Lipschitz polyhedral domain in Rd, showing, without extra regularity assumption, that the numerical solution converges to the exact solution both under discrete Hl-norms where 2 ≤ l ≤ m−1 and in H10(Ω). The strategy we applied consists of three steps: carrying a compactness argument to enumerate all limit points of uh, proving each of them identical to the unique solution to the PDE, and deducing the convergence. © 2025, American Institute of Mathematical Sciences. All rights reserved.

KW - C0 IP method

KW - compactness argument

KW - convergence analysis

KW - discontinuous Galerkin method

KW - mth Laplace equation

UR - http://www.scopus.com/inward/record.url?scp=105019257487&partnerID=8YFLogxK

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-105019257487&origin=recordpage

U2 - 10.3934/cac.2025021

DO - 10.3934/cac.2025021

M3 - RGC 21 - Publication in refereed journal

SN - 2837-0562

VL - 6

SP - 94

EP - 108

JO - Communications on Analysis and Computation

JF - Communications on Analysis and Computation

ER -