Finite element analysis of a nonlinear heat Equation with damping and pumping effects
Rishabh Shukla, Wasim Akram, Manil T. Mohan
TL;DR
This work analyzes a nonlinear heat equation with damping and pumping on bounded convex domains, establishing global well-posedness and regularity for a broad range of nonlinear exponents. It develops a unified finite element framework using conforming, nonconforming, and discontinuous Galerkin methods, providing a priori error estimates for semi- and fully discrete schemes. The analysis introduces projection-based tools (Ritz, Scott–Zhang, Clement, and $L^2$ projections) to relax restrictions on the damping exponent $p$ and verifies convergence with comprehensive numerical experiments in multiple dimensions. The results offer rigorous guidance for accurately approximating nonlinear damped–pumped reaction–diffusion systems with competing nonlinearities and have potential applications in simulations of phase transitions, pattern formation, and excitable media.
Abstract
We study the following nonlinear heat equation with damping and pumping effects (a reaction-diffusion equation) posed on a bounded simply connected convex domain $Ω\subset \mathbb{R}^d$, $d \geq 1$ with Lipschitz boundary $\partialΩ$: $$ \frac{\partial u(t)}{\partial t} - νΔu(t) + α|u(t)|^{p-2}u(t) - \sum_{\ell=1}^M β_{\ell} |u(t)|^{q_{\ell}-2}u(t) = f(t), \quad t>0, $$ subject to homogeneous Dirichlet boundary conditions and the initial condition $u(0)=u_0$, where $2 \leq p < \infty$ and $2 \leq q_{\ell} < p$ for $1 \leq \ell \leq M$. For $u_0 \in L^2(Ω)$ and $f \in L^2(0,T;H^{-1}(Ω))$, we establish the existence and uniqueness of a weak solution for all dimensions $d \in \mathbb{N}$ and damping exponents $2 \leq p < \infty$. Furthermore, for $u_0 \in H^2(Ω) \cap H_0^1(Ω)$ and $f \in H^1(0,T;H^1(Ω))$, we obtain regularity results: these hold for every $2 \leq p < \infty$ when $1 \leq d \leq 4$, and for $2 \leq p \leq \frac{2d-6}{d-4}$ when $d \geq 5$. We further conduct finite element analysis using conforming, nonconforming, and discontinuous Galerkin methods, deriving a priori error estimates for both semi- and fully discrete schemes, supported by numerical results. To relax restrictions on $p$ in the semidiscrete analysis, we use appropriate projection/interpolation operators: the Ritz projection in the conforming case ($2 \le p \le \frac{2d}{d-2}$), the Scott-Zhang interpolation for $\frac{2d}{d-2} < p \le \frac{2d-6}{d-4}$, the Clément interpolation in the nonconforming setting, and the $L^2$-projection in the DG framework. In the fully discrete case, error estimates hold for the above $p$-range under $u_0 \in D(A^{3/2})$ and $f \in H^1(0,T;H^1(Ω))$.