A direct solver for symmetric sparse matrices from finite element problems is presented. The solver is supposed to work as a local solver of domain decomposition methods for hybrid parallelization on cluster systems of multi-core CPUs, then it is required to run on shared memory computers and to have an ability of kernel detection. Symmetric pivoting with a given threshold factorizes a matrix with a decomposition introduced by a nested bisection and selects suspicious null pivots from the threshold. The Schur complement constructed from the suspicious null pivots is examined by a factorization with 1×1 and 2×2 pivoting and by a robust kernel detection algorithm based on measurements of residuals with orthogonal projections onto supposed image spaces. A static data structure from the nested bisection and a block sub- structure for Schur complements on all bisection-levels can well employ level 3 BLAS routines. Asynchronous task execution for each block can reduce idling time of processors drastically, then the solver has high parallel efficiency. Competitive performance of the developed solver to Intel Pardiso on shared memory computers is shown by numerical experiments.