Date:
Wed, 24/05/202314:00-15:00
Title: On new ideas for inequalities analysis: Application to error estimates of numerical approximations
Abstract: Even today, improving the accuracy of approximation remains a challenging problem in numerical analysis. Hence, the possibility to accurately estimate the upper bound which appears in error estimates remains an important issue. Given that this bound directly depends on the remainder which appears in Taylor’s formula, I will show in this talk how to derive a new Taylor-like formula getting an optimized reduced remainder. Then, I will consider two applications: the Lagrange interpolation error and the finite elements approximation error. The second problem I will analyze treats on the ability to evaluate the relative accuracy between two numerical methods. Here, I will consider the framework of finite elements method applied to the approximation of second order elliptic PDE’s solution. Based on a geometrical and probabilistic interpretation of Bramble Hilbert Lemma, I will derive several probabilistic laws to assess the relative accuracy between two Lagrange finite elements P_{k1} and P_{k2}, (k1 < k2). Additional results and problems will be discussed, especially the case where P_{k2} finite element becomes overqualified.
Abstract: Even today, improving the accuracy of approximation remains a challenging problem in numerical analysis. Hence, the possibility to accurately estimate the upper bound which appears in error estimates remains an important issue. Given that this bound directly depends on the remainder which appears in Taylor’s formula, I will show in this talk how to derive a new Taylor-like formula getting an optimized reduced remainder. Then, I will consider two applications: the Lagrange interpolation error and the finite elements approximation error. The second problem I will analyze treats on the ability to evaluate the relative accuracy between two numerical methods. Here, I will consider the framework of finite elements method applied to the approximation of second order elliptic PDE’s solution. Based on a geometrical and probabilistic interpretation of Bramble Hilbert Lemma, I will derive several probabilistic laws to assess the relative accuracy between two Lagrange finite elements P_{k1} and P_{k2}, (k1 < k2). Additional results and problems will be discussed, especially the case where P_{k2} finite element becomes overqualified.