The focus of the present work is the (theoretical) approximation of a solution of the p(x)-Poisson equation. To devise an iterative solver with guaranteed convergence, we will consider a relaxation of the original problem in terms of a truncation of the nonlinearity from below and from above by using a pair of positive cut-off parameters. We will then verify that, for any such pair, a damped Ka\v{c}anov scheme generates a sequence converging to a solution of the relaxed equation. Subsequently, it will be shown that the solutions of the relaxed problems converge to the solution of the original problem in the discrete setting. Finally, the discrete solutions of the unrelaxed problem converge to the continuous solution. Our work will finally be rounded up with some numerical experiments that underline the analytical findings.

翻译：本工作的重点是对p(x)-泊松方程的解进行（理论上的）逼近。为了设计一个具有保证收敛性的迭代求解器，我们将考虑原问题的松弛形式，即利用一对正截断参数从下方和上方对非线性项进行截断。随后，我们将证明，对于任意这样的一对参数，阻尼Kačanov方案生成的序列收敛到松弛方程的解。接着，将证明在离散设定下松弛问题的解收敛到原问题的解。最后，原问题的离散解将收敛到连续解。我们的工作最终将通过一些数值实验加以总结，这些实验验证了分析发现。