In this paper, we investigate the computational complexity of solutions to the Laplace and the diffusion equation. We show that for a certain class of initial-boundary value problems of the Laplace and the diffusion equation, the solution operator is $\# P_1/ \#P$-complete in the sense that it maps polynomial-time computable functions to the set of $\#P_1/ \#P$-complete functions. Consequently, there exists polynomial-time (Turing) computable input data such that the solution is not polynomial-time computable, unless $FP=\#P$ or $FP_1=\#P_1$. In this case, we can, in general, not simulate the solution of the Laplace or the diffusion equation on a digital computer without having a complexity blowup, i.e., the computation time for obtaining an approximation of the solution with up to a finite number of significant digits grows non-polynomially in the number of digits. This indicates that the computational complexity of the solution operator that models a physical phenomena is intrinsically high, independent of the numerical algorithm that is used to approximate a solution.

翻译：本文研究了拉普拉斯方程与扩散方程解的计算复杂度。我们证明，对于拉普拉斯方程和扩散方程的某类初边值问题，其解算子属于$\# P_1/ \#P$-完全问题，即它将多项式时间可计算函数映射到$\#P_1/ \#P$-完全函数集。因此，除非$FP=\#P$或$FP_1=\#P_1$，否则存在多项式时间（图灵）可计算的输入数据，使得其解无法在多项式时间内计算。在此情形下，我们通常无法在数字计算机上模拟拉普拉斯方程或扩散方程的解而不产生复杂性爆炸，即：获取具有有限有效数字近似解所需的计算时间随数字位数呈非多项式增长。这表明，用于刻画物理现象的算子解的计算复杂度本质上是极高的，且与用于近似解的数值算法无关。