Analysis and numerics for a parabolic equation with impulsive forcing Original Research Article

The paper considers a one-dimensional particle-continuum model, with impulsive interaction between the fluid and a number of pointwise particles. A simplification results in a system of ODEs coupled with a parabolic PDE forced by a nonlinear term involving a sum of Dirac delta functions. The existence of a mild solution is proved using a combination of energy estimates and semigroup theory. However, the regularity of these solutions is shown to be limited to C0,1 by the impulsive terms. The convergence of a Galerkin method is established simultaneously with a proof of continuous dependence, and thus uniqueness, of solutions for the underlying system. The peculiarities of the system imply this analysis must be performed in L∞. The C0,1 regularity of the solution determines a suboptimal rate of convergence for the Galerkin method. The theoretical results are verified by MATLAB computations.