Block implicit methods (BIM) have desirable stability properties, provide high-order of accuracy and no need to have multiple initial values. In a recent work, we introduced some A-stable BIM for parabolic problems and we will discuss how to these A-stable BIM to efficiently solve parabolic problems. This online discusison consists of two parts which will be hold in two different time.  In the first part of this discussion, we consider some L-stable BIM. Similar to Runge-Kutta methods, a tableau including two matrices and two vectors defines a particular BIM with the required order of accuracy and stability properties. More specifically, we consider L-stable BIM with a positive definite matrix and a positive diagonal matrix; both matrix properties are desirable but not available in Runge-Kutta methods.

In the second part of this discussion, we will show that the traditional finite element theory for parabolic problems discretized by the backward Euler or Crank-Nicolson schemes can also be extended to this family of BIM. To solve the resulting large sparse linear system of equations we will discuss several tensor structure preserving domain decomposition preconditioners for Krylov subspace methods.