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We introduce the multivariate decomposition finite element method (MDFEM) forsolving elliptic PDEs with uniform random diffusion coefficients. We show thatthe MDFEM can be used to reduce the computational complexity of estimating theexpected value of a linear functional of the solution of the PDE. The proposedalgorithm combines the multivariate decomposition method (MDM), to computeinfinite dimensional integrals, with the finite element method (FEM), to solvedifferent instances of the PDE. The strategy of the MDFEM is to decompose theinfinite-dimensional problem into multiple finite-dimensional ones which lendsitself to easier parallelization than to solve a single large dimensionalproblem. Our first result adjusts the analysis of the multivariatedecomposition method to incorporate the log-factor which typically appears inerror bounds for multivariate quadrature, i.e., cubature, methods; and we takecare of the fact that the number of points $n$ needs to come, e.g., in powersof 2 for higher order approximations. For the further analysis we specializethe cubature methods to be two types of quasi-Monte Carlo (QMC) rules, beingdigitally shifted polynomial lattice rules and interlaced polynomial latticerules. The second and main contribution then presents a bound on the error ofthe MDFEM and shows higher-order convergence w.r.t. the total computationalcost in case of the interlaced polynomial lattice rules in combination with ahigher-order finite element method.

MDFEM: Multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM