In this work a unifying approach is presented that leads to bounds for the distance in natural norms between solutions belonging to different spaces, of well-posed linear variational problems with the same input data. This is done in a general hilbertian framework, and in this sense, well-known inequalities such as Céa's or Babuška's for coercive and non coercive problems are extended and/or refined, as mere by-products of this unified setting. More particularly such an approach gives rise to both an improvement and a generalization to the weakly coercive case, of second Strang's inequality for abstract coercive problems. Additionally several aspects specific to linear variational problems are the subject of a thorough analysis beforehand, which also allows for clarifications and further refinements about the concept of weak coercivity.Key words: Babuška, Brezzi, Céa, Dupire, error bounds, linear, Strang, variational problems, weak coercivity. . Visiting professor in a . Corresponding authorEmail addresses: jacuminato@gmail.com, vitoriano.ruas@upmc.fr (Vitoriano RUAS). Preprint submitted to October 20, 2015Notation -0 E : the null element of a vector space E;-e E : the norm of an element e of a normed vector space E;-S E : the unit sphere of E (i.e. the set of elements e ∈ E such that e E = 1); -V ⊕ W : the direct sum of two subspaces V and W ;-A |V : the restriction to subspace V of X of operator A;-S : the closure of a set S in a normed vector space; -A * : the adjoint of an operator A;-E \ S : the set of elements in E that do not belong to its subset S.-L 2 (Ω) : the space of square (Lebesgue) integrable functions in a bounded open set Ω of N .-H m (Ω) : the subspace of L 2 (Ω) of functions having all derivatives up to the m-th order in L 2 (Ω).-C 0 (Ω) : the space of continuous functions in the closure of a bounded open set Ω ⊂ N .-C 1 (Ω) : the subspace of C 0 (Ω) of functions having all first order derivatives in C 0 (Ω).2

Summary.A family of simplicial finite element methods having the simplest possible structure, is introduced to sotve biharmonic problems in IR', n> 3, using the primal variable. The family is inspired in the MORLEY triangle for the two dimensional case, and in some sense this element can be viewed as its member corresponding to the value n = 2.