For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the Levi function can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the Levi function framework. The well-posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the Levi function, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme (ADS) for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method (DRM) which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2-dimensional and 3-dimensional cases are presented to show the validity of the proposed schemes.

翻译：对于描述非均匀介质中物理场分布的变系数椭圆方程边值问题，Levi函数可将解表示为体积势与面积势的组合，但其缺陷在于解表达式中的体积势计算成本高昂，且涉及密度函数对的积分方程可解性要求严格。本文在Levi函数框架下，针对散度形式椭圆方程的解提出一种改进的积分表达式。严格证明了待求密度函数所对应线性积分系统的适定性。基于Levi函数的奇性分解，我们提出两种处理体积积分的方案以实现密度函数的高效求解：其一是针对连续被积函数的自适应离散化方案（ADS），通过保证全域积分精度的一致性实现密度函数的高效计算；其二是对偶互易法（DRM），该无网格方法通过将体积密度表示为内部网格点确定的径向基函数组合，将体积积分等价转化为边界积分。数值实验验证了所提方案具有令人满意的计算成本。文中通过二维与三维算例展示了所提方法的有效性。