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 $\#P$-complete in the sense that it maps polynomial-time computable functions to the set of $\#P$-complete functions. Consequently, there exists polynomial-time (Turing) computable input data such that the solution is not polynomial-time computable, unless $FP=\#P$. 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 exponentially in the number of digits. This shows 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. This indicates that there is a fundamental problem in simulating physical phenomena on digital hardware.

翻译：在本文中, 我们调查 Laplace 解决方案的计算复杂性 和 扩散方程式 。 我们显示, 对于 Laplace 和 扩散方程式 的某类初始界限值问题, 解决方案操作员是完全的 $P$, 因为它映射了 $P$- 完成函数组的多元时间可计算函数。 因此, 存在多元时间( 试算) 可计算输入数据, 使得解决方案不是多元时间可计算, 除非$FP ⁇ P$ 。 在这种情况下, 一般来说, 我们无法在数字计算机上模拟 Laplace 解决方案或扩散方程式的解决方案, 而没有复杂的打击, 也就是说, 在数字硬件上模拟物理现象存在根本性的问题 。