Bit-Complexity of Solving Systems of Linear Evolutionary Partial Differential Equations

Finite Elements are a common method for solving differential equations via discretization. Under suitable hypotheses, the solution \(\mathbf {u}=\mathbf {u}(t,\vec x)\) of a well-posed initial/boundary-value problem for a linear evolutionary system of PDEs is approximated up to absolute error \(1/2^n\) by repeatedly (exponentially often in n) multiplying a matrix \(\mathbf {A}_n\) to the vector from the previous time step, starting with the initial condition \(\mathbf {u}(0)\), approximated by the spatial grid vector \(\mathbf {u}(0)_n\). The dimension of the matrix \(A_n\) is exponential in n, which is the number of the bits of the output.