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We consider the numerical approximation of second-order semi-linear parabolicstochastic partial differential equations interpreted in the mild sense whichwe solve on general two-dimensional domains with a $\mathcal{C}^2$ boundarywith homogeneous Dirichlet boundary conditions. The equations are driven byGaussian additive noise, and several Lipschitz-like conditions are imposed onthe nonlinear function. We discretize in space with a spectral Galerkin methodand in time using an explicit Euler-like scheme. For irregular shapes, thenecessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundaryintegral equation method. This yields a nonlinear eigenvalue problem, which isdiscretized using a boundary element collocation method and is solved with theBeyn contour integral algorithm. We present an error analysis as well asnumerical results on an exemplary asymmetric shape, and point out limitationsof the approach.

A spectral Galerkin exponential Euler time-stepping scheme for parabolic SPDEs on two-dimensional domains with a C2-boundary