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textabstractThe aim of this paper is to show that a high-order discretization can be used to improve the convergence of a multilevel\udMonte Carlo method for elliptic partial differential equations with lognormal random coefficients in combination with\udthe multigrid solution method. To demonstrate this, we consider a fourth-order accurate finite-volume discretization.\udWith the help of the Matérn family of covariance functions, we simulate the coefficient field with different degrees\udof smoothness. The idea behind using a fourth-order scheme is to capture the additional regularity in the solution\udintroduced due to higher smoothness of the random field. Second-order schemes previously utilized for these types of\udproblems are not able to fully exploit this additional regularity. We also propose a practical way of combining a full\udmultigrid solver with the multilevel Monte Carlo estimator constructed on the same mesh hierarchy. Through this\udintegration, one full multigrid solve at any level provides a valid sample for all the preceding Monte Carlo levels. The\udnumerical results show that the fourth-order multilevel estimator consistently outperforms the second-order variant.\udIn addition, we observe an asymptotic gain for the standard Monte Carlo estimator

A MULTIGRID MULTILEVEL MONTE CARLO METHOD USING HIGH-ORDER FINITE-VOLUME SCHEME FOR LOGNORMAL DIFFUSION PROBLEMS