Suppose a vector-valued diffusion process X satisfies the stochastic differential equation $$dX_t = b(X_t)dt + \sigma(X_t) dW_t,$$ in which $W$ is a Brownian motion and $b,\sigma$ are such that strong existence and uniqueness of solution to the SDE hold. Assume also that $X$ has a stationary distribution. Sufficient condition for the existence of a stationary distribution can be found in, e.g. Khasminskii, Stochastic Stability of Differential Equations. In a recent book by Mao and Yuan, Stochastic differential equations with Markovian switching, a Euler Mariamman method is proposed to numerically approximate the stationary distribution of $X$. But there is no mentioning on the efficiency of the algorithm, say, the convergence rate. What will be the convergence rate? And are there any other methods in the literature available? Many thanks for your comments and suggestions.

I don't have that book on me (I am saving up to get it!), but if you are mentioning this algorithm then it seems that the convergence rate is simply the convergence of the path to its stable distribution itself. Indeed, the proof in a nutshell is that the EM algorithm works for finding the steady state distribution is simply that the algorithm follows the actual path close enough to approximate the same distribution. Finding results about convergence rate and the like would be tough because it has adaptive timesteps and the like.

For an intuition on the rate of convergence of SDEs to their steady state distribution, you can look at Mao's SDE book in Chapter 5. It seems as though solutions converge in p-norms exponentially (this tends to be a general property of Markov processes), but I might be mistaken.

For details on why the algorithm works, Mao has a paper on Euler methods for SDEs with Markovian Switching. Here he shows that the EM method converges with order 1/2 in the $L^2$ sense. The key from the analysis is that the Higham, Mao, Stuart proof for SDEs pretty much carries over, so your intuition about Euler methods in SDEs with Markovian switching can pretty much follow your Euler method intuitions from SDEs. Again, this error is just the pathwise error and not the steady state error, but this estimate is required for the convergence to the steady state.

Lastly, if you need to know the error / want an algorithm with known convergence rates, you can directly solve the Fokker-Plank equation found in Theorem 2.4 of the first linked paper. Since you have a single Ito-dimension, its just a diffusion equation with drift (and an extra constraint).