We consider the numerical approximation of variational problems with orthotropic growth, that is those where the integrand depends strongly on the coordinate directions with possibly different growth in each direction. Under realistic regularity assumptions we derive optimal error estimates. These estimates depend on the existence of an orthotropically stable interpolation operator. Over certain meshes we construct an orthotropically stable interpolant that is also a projection. Numerical experiments illustrate and explore the limits of our theory.

翻译：本文考虑具有正交增长特性的变分问题的数值逼近，即被积函数强烈依赖于坐标方向且各方向可能具有不同增长速率的问题。在符合实际的正则性假设下，我们推导出最优误差估计。这些估计依赖于正交稳定插值算子的存在性。我们在特定网格上构造了一个满足正交稳定性的插值算子，该算子同时具有投影性质。数值实验验证并探索了所提出理论的适用范围。