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

翻译：我们考虑在温和解意义下解释的二阶半线性抛物型随机偏微分方程的数值逼近问题，并在具有$\mathcal{C}^2$边界的一般二维区域上求解，施加齐次狄利克雷边界条件。这些方程由高斯加性噪声驱动，并对非线性函数施加若干类利普希茨条件。我们采用谱Galerkin方法进行空间离散，并利用显式欧拉类格式进行时间离散。对于不规则形状，所需的狄利克雷特征值和特征函数通过边界积分方程方法获得。这产生一个非线性特征值问题，采用边界元配置法离散，并使用Beyn围道积分算法求解。我们给出了误差分析以及在示例性非对称形状上的数值结果，并指出了该方法的局限性。