We derive and analyze a fully computable discrete scheme for fractional partial differential equations posed on the full space $\mathbb{R}^d$ . Based on a reformulation using the well-known Caffarelli-Silvestre extension, we study a modified variational formulation to obtain well-posedness of the discrete problem. Our scheme is obtained by combining a diagonalization procedure with a reformulation using boundary integral equations and a coupling of finite elements and boundary elements. For our discrete method we present a-priori estimates as well as numerical examples.

翻译：我们推导并分析了一种针对全空间$\mathbb{R}^d$上分数阶偏微分方程的完全可计算离散格式。基于著名的Caffarelli-Silvestre延拓的重新表述，我们研究了一种修正的变分形式，以确保离散问题的适定性。该格式通过结合对角化过程与基于边界积分方程的重新表述，以及有限元与边界元的耦合而获得。针对所提出的离散方法，我们给出了先验估计和数值算例。