A singularly perturbed reaction–diffusion problem posed on the unit square in \(\mathbb {R}^2\) is considered. To solve this problem numerically, a finite volume method (FVM) whose primal mesh is Shishkin is constructed; the FVM solution is piecewise bilinear on this mesh. Working in the standard energy norm, a superclose result (for the difference between the FVM solution and the Lagrange interpolant of the exact solution) is derived. This result yields an improved bound for the \(L^2\) error of the FVM solution, and implies that a simple postprocessing of the FVM solution produces (in the energy norm) a higher-order approximation of the true solution. Next, we analyse errors in a balanced norm that is stronger than the energy norm; using a more complicated approximant of the exact solution from our piecewise bilinear space, we prove an optimal-order error bound and an associated supercloseness result showing that the difference between the FVM solution and our approximant is of higher order than the error itself. Finally, numerical experiments demonstrate the sharpness of our error bounds.

考虑了\(\mathbb {R}^2\)中单位正方形上提出的奇异扰动反应扩散问题。为了数值求解该问题，构造了原始网格为Shishkin的有限体积法（FVM）；FVM 解在此网格上是分段双线性的。在标准能量范数下工作，得出超接近结果（FVM 解与精确解的拉格朗日插值之间的差异）。这个结果产生了\(L^2\)的改进界限FVM 解的误差，并且意味着 FVM 解的简单后处理会产生（在能量范数中）真实解的高阶近似。接下来，我们分析比能量范数更强的平衡范数中的误差；使用来自分段双线性空间的精确解的更复杂的近似，我们证明了最优阶误差界限和相关的超接近结果，表明 FVM 解和我们的近似之间的差异比误差本身具有更高的阶数。最后，数值实验证明了我们的误差范围的锐度。