Finite element analysis of a nonlinear heat Equation with damping and pumping effects
Authors
Rishabh Shukla
Wasim Akram
Manil T. Mohan
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 $f \in H^1(0,T;H^1(Ω))$.
