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Conformal Finite Element Methods for Nonlinear Rosenau-Burgers-Biharmonic Models

Ankur, Ram Jiwari, Akil Narayan

TL;DR

We address the numerical approximation of the nonlinear Rosenau–Burgers–biharmonic model, a time-dependent fourth-order PDE with a biharmonic term in u_t. The authors develop and analyze two finite element formulations: a primal formulation P1 in H_0^2(Ω) and a mixed formulation P2 with p = -Δu, enabling robust stability and convergence proofs in Bochner spaces. They establish well-posedness, semidiscrete error estimates, and fully discrete schemes (backward Euler) for both formulations, highlighting that the mixed approach requires less regularity and is easier to implement, while achieving optimal convergence in L^2, H^1, and H^2 norms. Numerical experiments in 2D using C^0 Taylor–Hood elements confirm the theoretical rates on several domains and boundary conditions, underscoring the practical viability of the proposed methods for Rosenau-type dispersive PDEs.

Abstract

We present a novel and comparative analysis of finite element discretizations for a nonlinear Rosenau-Burgers model including a biharmonic term. We analyze both continuous and mixed finite element approaches, providing stability, existence, and uniqueness statements of the corresponding variational methods. We also obtain optimal error estimates of the semidiscrete scheme in corresponding Bôchner spaces. Finally, we construct a fully discrete scheme through a backward Euler discretization of the time derivative, and prove well-posedness statements for this fully discrete scheme. Our findings show that the mixed approach removes some theoretical impediments to analysis and is numerically easier to implement. We provide numerical simulations for the mixed formulation approach using $C^0$ Taylor-Hood finite elements on several domains. Our numerical results confirm that the algorithm has optimal convergence in accordance with the observed theoretical results.

Conformal Finite Element Methods for Nonlinear Rosenau-Burgers-Biharmonic Models

TL;DR

We address the numerical approximation of the nonlinear Rosenau–Burgers–biharmonic model, a time-dependent fourth-order PDE with a biharmonic term in u_t. The authors develop and analyze two finite element formulations: a primal formulation P1 in H_0^2(Ω) and a mixed formulation P2 with p = -Δu, enabling robust stability and convergence proofs in Bochner spaces. They establish well-posedness, semidiscrete error estimates, and fully discrete schemes (backward Euler) for both formulations, highlighting that the mixed approach requires less regularity and is easier to implement, while achieving optimal convergence in L^2, H^1, and H^2 norms. Numerical experiments in 2D using C^0 Taylor–Hood elements confirm the theoretical rates on several domains and boundary conditions, underscoring the practical viability of the proposed methods for Rosenau-type dispersive PDEs.

Abstract

We present a novel and comparative analysis of finite element discretizations for a nonlinear Rosenau-Burgers model including a biharmonic term. We analyze both continuous and mixed finite element approaches, providing stability, existence, and uniqueness statements of the corresponding variational methods. We also obtain optimal error estimates of the semidiscrete scheme in corresponding Bôchner spaces. Finally, we construct a fully discrete scheme through a backward Euler discretization of the time derivative, and prove well-posedness statements for this fully discrete scheme. Our findings show that the mixed approach removes some theoretical impediments to analysis and is numerically easier to implement. We provide numerical simulations for the mixed formulation approach using Taylor-Hood finite elements on several domains. Our numerical results confirm that the algorithm has optimal convergence in accordance with the observed theoretical results.
Paper Structure (23 sections, 18 theorems, 175 equations, 4 figures, 1 table)

This paper contains 23 sections, 18 theorems, 175 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. A brief literature review
  3. Contribution
  4. Structure of the paper
  5. Problem setup and weak formulations
  6. Notation and assumptions
  7. The weak formulations
  8. Problem $P^1$:
  9. Problem $P^2$:
  10. Analysis of problem $P^1$
  11. Well-posedness of the variational problem
  12. Semidiscrete Error Estimates
  13. Preliminary lemmas
  14. Semidiscrete scheme convergence
  15. Full Discretization of Problem $P^1$
  16. ...and 8 more sections

Key Result

Lemma 3

The function $u \mapsto \|u\| + \|\Delta u\|$ is a norm on $H^2(\Omega)$ and is equivalent to $\|u\|_2$.

Figures (4)

  • Figure 1: Six different domains $\Omega_i$, $i \in [6]$, considered in the examples.
  • Figure 2: \ref{['exx2']} plots of the numerical solution at time $T=1$ with $h=1/16,k=0.001$ using ${P_2}\times P_1$ elements. Columns moving left to right: $\beta = 1, 0.5, 0.1$, and $0.01$. Rows: Case-I (top) and Case-II (bottom).
  • Figure 3: \ref{['exm2']} with ${P_2}\times P_1$ elements and $k = 0.001$, $T = 1$: Surface (top) and error profiles (middle) of the numerical solutions on domains $\Omega_1$, $\Omega_3$, and $\Omega_4$ with $h=1/16$. Bottom left: $L^2$ errors showing a numerical convergence rate of $h^3$ (red) relative to ideal $h^3$ convergence (blue). Bottom right: Computed orders of convergence in the $H^1$ and $H^2$ norms.
  • Figure 4: \ref{['exm4']} with ${P_2}\times P_1$ elements and $k = 0.001$, $T = 1$: Surface (top) and error profiles (middle) of the numerical solutions on domains $\Omega_2$, $\Omega_5$, and $\Omega_6$ with $h=1/16$. Bottom left: $L^2$ errors showing a numerical convergence rate of $h^3$ (red) relative to ideal $h^3$ convergence (blue). Bottom middle: $H^1$ errors showing a numerical convergence rate of $h^2$ along with a tabulation CPU times required to compute the solutions. Bottom right: $H^2$ errors showing a numerical convergence rate of $h$ (red) relative to ideal $h$ convergence (blue).

Theorems & Definitions (39)

  • Lemma 3
  • Lemma 5: Stability estimate
  • proof
  • Theorem 6: Problem $P^1$ existence and uniqueness
  • proof
  • Lemma 7
  • proof
  • Lemma 8: Stability estimate
  • proof
  • Theorem 9
  • ...and 29 more